| L(s) = 1 | − 8·5-s − 16·19-s − 16·23-s + 28·25-s − 24·29-s − 8·43-s + 8·47-s − 4·49-s − 8·67-s − 32·71-s + 8·73-s + 128·95-s − 32·97-s − 16·101-s + 128·115-s + 44·121-s − 80·125-s + 127-s + 131-s + 137-s + 139-s + 192·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.57·5-s − 3.67·19-s − 3.33·23-s + 28/5·25-s − 4.45·29-s − 1.21·43-s + 1.16·47-s − 4/7·49-s − 0.977·67-s − 3.79·71-s + 0.936·73-s + 13.1·95-s − 3.24·97-s − 1.59·101-s + 11.9·115-s + 4·121-s − 7.15·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 15.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0001875811816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0001875811816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( ( 1 + 4 T + 2 p T^{2} + 32 T^{3} + 87 T^{4} + 32 p T^{5} + 2 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 4 p T^{2} + 994 T^{4} - 15968 T^{6} + 198763 T^{8} - 15968 p^{2} T^{10} + 994 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 8 T^{2} + 228 T^{4} - 4504 T^{6} + 20006 T^{8} - 4504 p^{2} T^{10} + 228 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \) |
| 19 | \( ( 1 + 8 T + 30 T^{2} + 112 T^{3} + 611 T^{4} + 112 p T^{5} + 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 8 T + 2 p T^{2} + 88 T^{3} + 231 T^{4} + 88 p T^{5} + 2 p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( 1 - 156 T^{2} + 12106 T^{4} - 610416 T^{6} + 22035219 T^{8} - 610416 p^{2} T^{10} + 12106 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 - 164 T^{2} + 14154 T^{4} - 829168 T^{6} + 35617331 T^{8} - 829168 p^{2} T^{10} + 14154 p^{4} T^{12} - 164 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 - 116 T^{2} + 11362 T^{4} - 660608 T^{6} + 32960203 T^{8} - 660608 p^{2} T^{10} + 11362 p^{4} T^{12} - 116 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 4 T + 84 T^{2} + 44 T^{3} + 3002 T^{4} + 44 p T^{5} + 84 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4 T + 76 T^{2} - 44 T^{3} + 2154 T^{4} - 44 p T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 124 T^{2} - 192 T^{3} + 7734 T^{4} - 192 p T^{5} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 232 T^{2} + 28836 T^{4} - 2582456 T^{6} + 175948646 T^{8} - 2582456 p^{2} T^{10} + 28836 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 92 T^{2} + 6486 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 4 T + 228 T^{2} + 620 T^{3} + 21530 T^{4} + 620 p T^{5} + 228 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 16 T + 310 T^{2} + 3320 T^{3} + 33807 T^{4} + 3320 p T^{5} + 310 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 4 T + 156 T^{2} - 20 p T^{3} + 11402 T^{4} - 20 p^{2} T^{5} + 156 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 248 T^{2} + 37188 T^{4} - 3949096 T^{6} + 344639558 T^{8} - 3949096 p^{2} T^{10} + 37188 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 232 T^{2} + 34596 T^{4} - 3949112 T^{6} + 355979174 T^{8} - 3949112 p^{2} T^{10} + 34596 p^{4} T^{12} - 232 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 292 T^{2} + 47890 T^{4} - 6052768 T^{6} + 614802619 T^{8} - 6052768 p^{2} T^{10} + 47890 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 16 T + 300 T^{2} + 4208 T^{3} + 41318 T^{4} + 4208 p T^{5} + 300 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.27474327990519760450708067833, −3.26480054458226625485971338309, −3.15564696526782669292391451012, −3.12697509011909574862885323901, −3.00063672902453862189961085630, −2.77813460786439712269812502144, −2.52949190664231381423286323178, −2.43282578213911629451118998684, −2.27117824727112993379756188076, −2.20273953877551607169556194303, −2.15572453378898266701512831667, −2.12905834446690942954848559879, −2.07593939814274949919371839345, −1.72927384573438489140861982405, −1.70260187352271439341179710699, −1.56689526118957483960824107875, −1.48161806689713880112171711063, −1.18055839595837711451397901288, −1.17025935919739824696102865919, −0.913878713247935390473616394579, −0.78159452040336649814012626570, −0.28270229787612114910157787252, −0.25226567411892450093923872965, −0.07379981985015398068590179461, −0.01796108317131565239751983364,
0.01796108317131565239751983364, 0.07379981985015398068590179461, 0.25226567411892450093923872965, 0.28270229787612114910157787252, 0.78159452040336649814012626570, 0.913878713247935390473616394579, 1.17025935919739824696102865919, 1.18055839595837711451397901288, 1.48161806689713880112171711063, 1.56689526118957483960824107875, 1.70260187352271439341179710699, 1.72927384573438489140861982405, 2.07593939814274949919371839345, 2.12905834446690942954848559879, 2.15572453378898266701512831667, 2.20273953877551607169556194303, 2.27117824727112993379756188076, 2.43282578213911629451118998684, 2.52949190664231381423286323178, 2.77813460786439712269812502144, 3.00063672902453862189961085630, 3.12697509011909574862885323901, 3.15564696526782669292391451012, 3.26480054458226625485971338309, 3.27474327990519760450708067833
Plot not available for L-functions of degree greater than 10.
