| L(s) = 1 | + 12·9-s + 96·29-s − 240·41-s − 104·49-s + 112·61-s + 90·81-s − 48·89-s − 96·101-s − 560·109-s + 680·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 376·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 3.31·29-s − 5.85·41-s − 2.12·49-s + 1.83·61-s + 10/9·81-s − 0.539·89-s − 0.950·101-s − 5.13·109-s + 5.61·121-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 + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.22·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.195774024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.195774024\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - p T^{2} )^{4} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 52 T^{2} + 2598 T^{4} + 52 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 340 T^{2} + 55302 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 188 T^{2} + 19878 T^{4} - 188 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 508 T^{2} + 127878 T^{4} - 508 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 868 T^{2} + 402918 T^{4} - 868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 100 T^{2} + 377862 T^{4} + 100 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 24 T + 1646 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 964 T^{2} + 927366 T^{4} - 964 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 1436 T^{2} + 1671846 T^{4} - 1436 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 30 T + p^{2} T^{2} )^{8} \) |
| 43 | \( ( 1 + 5380 T^{2} + 13889382 T^{4} + 5380 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 2212 T^{2} + 8033478 T^{4} + 2212 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 6844 T^{2} + 23758566 T^{4} - 6844 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 620 T^{2} + 23982342 T^{4} + 620 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 28 T + 1158 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 188 T^{2} - 19408602 T^{4} - 188 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 8060 T^{2} + 10662 p^{2} T^{4} + 8060 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 19868 T^{2} + 155469318 T^{4} - 19868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 12868 T^{2} + 113720838 T^{4} - 12868 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 19780 T^{2} + 177798822 T^{4} + 19780 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 12 T + 15158 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 18622 T^{2} + p^{4} T^{4} )^{4} \) |
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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
−3.94128018751586063005240524035, −3.93614320539580547373888038785, −3.83086820180611660388973152491, −3.34556540200990599299587614637, −3.34458848949535101729350721092, −3.22506148017476563745953051824, −3.08070486439072699278584118564, −3.03314085395460581826555750573, −2.99427310682765133918668992403, −2.97495198543684524672578927369, −2.59726919221707569837774126124, −2.39570601185385423795769577560, −2.12746411040245758342997792029, −2.10650375247207722654224839821, −2.03942632449354422859508479873, −1.72655316081849404828273297077, −1.63397701155685421505631321437, −1.59482684231874935372457454869, −1.32915216554426906968797150615, −1.20556108502432135796861892208, −0.961790382652050355945873835789, −0.74958941260326331138937173134, −0.60092337457414300810580515099, −0.34527270164364590863104592076, −0.10343257890871128756177153181,
0.10343257890871128756177153181, 0.34527270164364590863104592076, 0.60092337457414300810580515099, 0.74958941260326331138937173134, 0.961790382652050355945873835789, 1.20556108502432135796861892208, 1.32915216554426906968797150615, 1.59482684231874935372457454869, 1.63397701155685421505631321437, 1.72655316081849404828273297077, 2.03942632449354422859508479873, 2.10650375247207722654224839821, 2.12746411040245758342997792029, 2.39570601185385423795769577560, 2.59726919221707569837774126124, 2.97495198543684524672578927369, 2.99427310682765133918668992403, 3.03314085395460581826555750573, 3.08070486439072699278584118564, 3.22506148017476563745953051824, 3.34458848949535101729350721092, 3.34556540200990599299587614637, 3.83086820180611660388973152491, 3.93614320539580547373888038785, 3.94128018751586063005240524035
Plot not available for L-functions of degree greater than 10.
