| L(s) = 1 | − 3-s + 4-s + 0.517·5-s + 7-s + 9-s − 1.93·11-s − 12-s + 1.73·13-s − 0.517·15-s + 16-s + 1.41·17-s + 0.517·20-s − 21-s − 0.732·25-s − 27-s + 28-s − 1.41·29-s + 1.93·33-s + 0.517·35-s + 36-s − 1.73·39-s + 1.93·41-s − 1.93·44-s + 0.517·45-s − 48-s − 1.41·51-s + 1.73·52-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 0.517·5-s + 7-s + 9-s − 1.93·11-s − 12-s + 1.73·13-s − 0.517·15-s + 16-s + 1.41·17-s + 0.517·20-s − 21-s − 0.732·25-s − 27-s + 28-s − 1.41·29-s + 1.93·33-s + 0.517·35-s + 36-s − 1.73·39-s + 1.93·41-s − 1.93·44-s + 0.517·45-s − 48-s − 1.41·51-s + 1.73·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.530705110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.530705110\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 1297 | \( 1 + T \) |
| good | 2 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.517T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + 1.93T + T^{2} \) |
| 13 | \( 1 - 1.73T + T^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.93T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 + 1.73T + T^{2} \) |
| 67 | \( 1 + 1.73T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.517T + T^{2} \) |
| 89 | \( 1 + 1.93T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.378550331570780564433528553181, −7.67892158602701574421415072372, −7.38012194847607403518738366387, −6.08765989855090536381342941422, −5.73506298856109246962553118443, −5.31092611731364447331794751752, −4.16050675858512346041414621405, −3.08896125901124138345973133422, −1.97174844217297154998158521474, −1.23481390897200077080695284226,
1.23481390897200077080695284226, 1.97174844217297154998158521474, 3.08896125901124138345973133422, 4.16050675858512346041414621405, 5.31092611731364447331794751752, 5.73506298856109246962553118443, 6.08765989855090536381342941422, 7.38012194847607403518738366387, 7.67892158602701574421415072372, 8.378550331570780564433528553181
