| L(s) = 1 | + 24·9-s − 160·17-s − 184·25-s − 256·41-s − 56·49-s + 224·73-s + 96·81-s + 1.02e3·89-s + 128·97-s + 272·113-s + 800·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 3.84e3·153-s + 157-s + 163-s + 167-s − 888·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 8/3·9-s − 9.41·17-s − 7.35·25-s − 6.24·41-s − 8/7·49-s + 3.06·73-s + 1.18·81-s + 11.5·89-s + 1.31·97-s + 2.40·113-s + 6.61·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 25.0·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 5.25·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3010044109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3010044109\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{8} \) |
| good | 3 | \( ( 1 - 4 p T^{2} + 56 p T^{4} - 2212 T^{6} + 17038 T^{8} - 2212 p^{4} T^{10} + 56 p^{9} T^{12} - 4 p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| 5 | \( ( 1 + 92 T^{2} + 5464 T^{4} + 211956 T^{6} + 6231214 T^{8} + 211956 p^{4} T^{10} + 5464 p^{8} T^{12} + 92 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 11 | \( ( 1 - 400 T^{2} + 47836 T^{4} + 1686928 T^{6} - 819692794 T^{8} + 1686928 p^{4} T^{10} + 47836 p^{8} T^{12} - 400 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 13 | \( ( 1 + 444 T^{2} + 142936 T^{4} + 35361044 T^{6} + 6575436334 T^{8} + 35361044 p^{4} T^{10} + 142936 p^{8} T^{12} + 444 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 17 | \( ( 1 + 40 T + 1308 T^{2} + 31512 T^{3} + 588230 T^{4} + 31512 p^{2} T^{5} + 1308 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 19 | \( ( 1 - 2636 T^{2} + 3120040 T^{4} - 2166005604 T^{6} + 964702997902 T^{8} - 2166005604 p^{4} T^{10} + 3120040 p^{8} T^{12} - 2636 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 23 | \( ( 1 - 1744 T^{2} + 1272156 T^{4} - 412076080 T^{6} + 99307893702 T^{8} - 412076080 p^{4} T^{10} + 1272156 p^{8} T^{12} - 1744 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 29 | \( ( 1 + 3384 T^{2} + 6555580 T^{4} + 8754768776 T^{6} + 8490907402822 T^{8} + 8754768776 p^{4} T^{10} + 6555580 p^{8} T^{12} + 3384 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 31 | \( ( 1 - 3944 T^{2} + 8438620 T^{4} - 12447428312 T^{6} + 13694235978694 T^{8} - 12447428312 p^{4} T^{10} + 8438620 p^{8} T^{12} - 3944 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 37 | \( ( 1 + 3512 T^{2} + 9188668 T^{4} + 18622781448 T^{6} + 27544347275206 T^{8} + 18622781448 p^{4} T^{10} + 9188668 p^{8} T^{12} + 3512 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 + 64 T + 4956 T^{2} + 221760 T^{3} + 11848326 T^{4} + 221760 p^{2} T^{5} + 4956 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 43 | \( ( 1 - 9360 T^{2} + 42404572 T^{4} - 124584286064 T^{6} + 265774122344710 T^{8} - 124584286064 p^{4} T^{10} + 42404572 p^{8} T^{12} - 9360 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 47 | \( ( 1 - 8392 T^{2} + 39566748 T^{4} - 127207295352 T^{6} + 316693927920198 T^{8} - 127207295352 p^{4} T^{10} + 39566748 p^{8} T^{12} - 8392 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 53 | \( ( 1 + 18920 T^{2} + 162796828 T^{4} + 840091728600 T^{6} + 2864724835962118 T^{8} + 840091728600 p^{4} T^{10} + 162796828 p^{8} T^{12} + 18920 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 - 1804 T^{2} - 12929112 T^{4} + 2548527324 T^{6} + 280743846121998 T^{8} + 2548527324 p^{4} T^{10} - 12929112 p^{8} T^{12} - 1804 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 + 16316 T^{2} + 140172120 T^{4} + 816942037524 T^{6} + 3499102878259502 T^{8} + 816942037524 p^{4} T^{10} + 140172120 p^{8} T^{12} + 16316 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 67 | \( ( 1 - 21344 T^{2} + 191426940 T^{4} - 1006794243744 T^{6} + 4374042881960006 T^{8} - 1006794243744 p^{4} T^{10} + 191426940 p^{8} T^{12} - 21344 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 - 9864 T^{2} + 51888284 T^{4} - 294531431096 T^{6} + 1789421441990854 T^{8} - 294531431096 p^{4} T^{10} + 51888284 p^{8} T^{12} - 9864 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 73 | \( ( 1 - 56 T + 18460 T^{2} - 736008 T^{3} + 138223494 T^{4} - 736008 p^{2} T^{5} + 18460 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 79 | \( ( 1 - 24968 T^{2} + 330869788 T^{4} - 3106127956152 T^{6} + 22189846569597766 T^{8} - 3106127956152 p^{4} T^{10} + 330869788 p^{8} T^{12} - 24968 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 83 | \( ( 1 - 31660 T^{2} + 529313064 T^{4} - 5903046467332 T^{6} + 47298953549416590 T^{8} - 5903046467332 p^{4} T^{10} + 529313064 p^{8} T^{12} - 31660 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 89 | \( ( 1 - 256 T + 48252 T^{2} - 6269952 T^{3} + 638304966 T^{4} - 6269952 p^{2} T^{5} + 48252 p^{4} T^{6} - 256 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 97 | \( ( 1 - 32 T + 19484 T^{2} - 1437536 T^{3} + 199130566 T^{4} - 1437536 p^{2} T^{5} + 19484 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.13533556105554834733756543527, −1.97452111142445493853105918580, −1.97135286501441075677859031835, −1.90564220425902136970039526023, −1.77003745856292977657923066914, −1.74548643153941500544293985029, −1.64469119432552745456114773721, −1.62245959057119261244762142305, −1.57423496113320449270195715648, −1.57176568855703879662529472895, −1.56703471558898591861223776748, −1.54078686473574330440114678746, −1.18237287691155943251167977677, −1.12201095996524681783516575140, −0.826900834475725654754600704480, −0.797906642111300974031887029089, −0.77815536135788859313925710042, −0.64697284539292249597475507311, −0.63418125394111519027939176801, −0.48159348919847393421940213583, −0.44562539874739773700159323587, −0.20461327754947243875761574581, −0.20150715140569849347525084978, −0.12295309318669135555512692106, −0.06589966824175060890681003402,
0.06589966824175060890681003402, 0.12295309318669135555512692106, 0.20150715140569849347525084978, 0.20461327754947243875761574581, 0.44562539874739773700159323587, 0.48159348919847393421940213583, 0.63418125394111519027939176801, 0.64697284539292249597475507311, 0.77815536135788859313925710042, 0.797906642111300974031887029089, 0.826900834475725654754600704480, 1.12201095996524681783516575140, 1.18237287691155943251167977677, 1.54078686473574330440114678746, 1.56703471558898591861223776748, 1.57176568855703879662529472895, 1.57423496113320449270195715648, 1.62245959057119261244762142305, 1.64469119432552745456114773721, 1.74548643153941500544293985029, 1.77003745856292977657923066914, 1.90564220425902136970039526023, 1.97135286501441075677859031835, 1.97452111142445493853105918580, 2.13533556105554834733756543527
Plot not available for L-functions of degree greater than 10.
