| L(s) = 1 | − 4·4-s − 12·9-s + 36·11-s + 4·16-s + 80·19-s − 36·29-s + 76·31-s + 48·36-s − 252·41-s − 144·44-s − 86·49-s − 252·59-s + 124·61-s + 16·64-s − 320·76-s − 28·79-s − 54·81-s − 432·99-s − 36·101-s + 128·109-s + 144·116-s + 254·121-s − 304·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4-s − 4/3·9-s + 3.27·11-s + 1/4·16-s + 4.21·19-s − 1.24·29-s + 2.45·31-s + 4/3·36-s − 6.14·41-s − 3.27·44-s − 1.75·49-s − 4.27·59-s + 2.03·61-s + 1/4·64-s − 4.21·76-s − 0.354·79-s − 2/3·81-s − 4.36·99-s − 0.356·101-s + 1.17·109-s + 1.24·116-s + 2.09·121-s − 2.45·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.209325169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209325169\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( ( 1 + 2 p T^{2} + p^{4} T^{4} )^{2} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 86 T^{2} + 961 T^{4} + 140438 T^{6} + 17886628 T^{8} + 140438 p^{4} T^{10} + 961 p^{8} T^{12} + 86 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 18 T + 359 T^{2} - 4518 T^{3} + 61428 T^{4} - 4518 p^{2} T^{5} + 359 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 + 194 T^{2} - 7295 T^{4} - 2365054 T^{6} + 128142244 T^{8} - 2365054 p^{4} T^{10} - 7295 p^{8} T^{12} + 194 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 + 796 T^{2} + 294342 T^{4} + 796 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 20 T + 606 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 - 2026 T^{2} + 2520769 T^{4} - 2075079850 T^{6} + 1281793762852 T^{8} - 2075079850 p^{4} T^{10} + 2520769 p^{8} T^{12} - 2026 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 + 18 T + 1745 T^{2} + 29466 T^{3} + 2063316 T^{4} + 29466 p^{2} T^{5} + 1745 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 38 T - 353 T^{2} + 4750 T^{3} + 918004 T^{4} + 4750 p^{2} T^{5} - 353 p^{4} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2996 T^{2} + 5107590 T^{4} - 2996 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 126 T + 9329 T^{2} + 508662 T^{3} + 22367460 T^{4} + 508662 p^{2} T^{5} + 9329 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 + 5366 T^{2} + 15786241 T^{4} + 17906342 p^{2} T^{6} + 17278468 p^{4} T^{8} + 17906342 p^{6} T^{10} + 15786241 p^{8} T^{12} + 5366 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 6586 T^{2} + 23629489 T^{4} - 65771385370 T^{6} + 154431460862212 T^{8} - 65771385370 p^{4} T^{10} + 23629489 p^{8} T^{12} - 6586 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 2236 T^{2} - 2409114 T^{4} + 2236 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 126 T + 10535 T^{2} + 660618 T^{3} + 33793140 T^{4} + 660618 p^{2} T^{5} + 10535 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 62 T - 2615 T^{2} + 60946 T^{3} + 13569316 T^{4} + 60946 p^{2} T^{5} - 2615 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 + 11366 T^{2} + 62047921 T^{4} + 305015623238 T^{6} + 1498206235247908 T^{8} + 305015623238 p^{4} T^{10} + 62047921 p^{8} T^{12} + 11366 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( ( 1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 12020 T^{2} + 71890278 T^{4} - 12020 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 14 T - 10985 T^{2} - 18214 T^{3} + 84841444 T^{4} - 18214 p^{2} T^{5} - 10985 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 2842 T^{2} - 67426319 T^{4} + 55172766278 T^{6} + 3065134361421604 T^{8} + 55172766278 p^{4} T^{10} - 67426319 p^{8} T^{12} - 2842 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( 1 + 16370 T^{2} + 25998577 T^{4} + 1062736487570 T^{6} + 26992272925906468 T^{8} + 1062736487570 p^{4} T^{10} + 25998577 p^{8} T^{12} + 16370 p^{12} T^{14} + p^{16} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.72338207841618927387152454125, −4.51003586083080936321769476797, −4.36662221632345458018163760801, −4.24728871458782374114736838911, −4.12605277536830009565110871454, −3.96931963037255981826496132611, −3.49273749827214638846403278300, −3.48362747418036868806323398673, −3.42765035755847002282668295940, −3.31053858385266271441226338778, −3.30135069558009102085854588276, −3.17004761265191816869436842429, −2.90885602185162006087620277845, −2.77310529942830853464972971226, −2.62919199190896191070047189822, −2.04016590131700430580857817992, −1.94646285531932939626986162584, −1.82362787747323728314929743744, −1.60478887184860539062542063932, −1.48096345091695642199033970694, −1.18692933468957815518892513413, −0.979639361914583887610375072374, −0.889725029346254656038137356788, −0.43419855552633868699859900531, −0.12480811659970975557644551784,
0.12480811659970975557644551784, 0.43419855552633868699859900531, 0.889725029346254656038137356788, 0.979639361914583887610375072374, 1.18692933468957815518892513413, 1.48096345091695642199033970694, 1.60478887184860539062542063932, 1.82362787747323728314929743744, 1.94646285531932939626986162584, 2.04016590131700430580857817992, 2.62919199190896191070047189822, 2.77310529942830853464972971226, 2.90885602185162006087620277845, 3.17004761265191816869436842429, 3.30135069558009102085854588276, 3.31053858385266271441226338778, 3.42765035755847002282668295940, 3.48362747418036868806323398673, 3.49273749827214638846403278300, 3.96931963037255981826496132611, 4.12605277536830009565110871454, 4.24728871458782374114736838911, 4.36662221632345458018163760801, 4.51003586083080936321769476797, 4.72338207841618927387152454125
Plot not available for L-functions of degree greater than 10.
