| L(s) = 1 | + 2·5-s − 3·9-s + 3·11-s − 3·13-s + 2·17-s − 19-s + 3·23-s − 25-s + 6·29-s − 3·37-s + 11·41-s − 43-s − 6·45-s − 3·47-s + 5·53-s + 6·55-s + 12·59-s − 15·61-s − 6·65-s + 67-s − 71-s − 3·73-s − 7·79-s + 9·81-s − 11·83-s + 4·85-s + 5·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 9-s + 0.904·11-s − 0.832·13-s + 0.485·17-s − 0.229·19-s + 0.625·23-s − 1/5·25-s + 1.11·29-s − 0.493·37-s + 1.71·41-s − 0.152·43-s − 0.894·45-s − 0.437·47-s + 0.686·53-s + 0.809·55-s + 1.56·59-s − 1.92·61-s − 0.744·65-s + 0.122·67-s − 0.118·71-s − 0.351·73-s − 0.787·79-s + 81-s − 1.20·83-s + 0.433·85-s + 0.529·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.76132649632927, −12.20101336393215, −11.76353414654895, −11.49398272276043, −10.89337872963487, −10.34619374083921, −10.02501891366051, −9.533244620321351, −9.030618235171836, −8.803125193866533, −8.228565837706977, −7.612332898310155, −7.243140286960298, −6.511275447924556, −6.308621494761844, −5.711684968645766, −5.352792845558891, −4.816639099982080, −4.249396552701824, −3.688504528690052, −2.958296909772255, −2.654402531498760, −2.076938815803003, −1.402776427865679, −0.8341289149581990, 0,
0.8341289149581990, 1.402776427865679, 2.076938815803003, 2.654402531498760, 2.958296909772255, 3.688504528690052, 4.249396552701824, 4.816639099982080, 5.352792845558891, 5.711684968645766, 6.308621494761844, 6.511275447924556, 7.243140286960298, 7.612332898310155, 8.228565837706977, 8.803125193866533, 9.030618235171836, 9.533244620321351, 10.02501891366051, 10.34619374083921, 10.89337872963487, 11.49398272276043, 11.76353414654895, 12.20101336393215, 12.76132649632927
