| L(s) = 1 | − 0.294·3-s − 3.39·5-s + 0.817·7-s − 2.91·9-s − 1.09·11-s − 1.08·13-s + 15-s + 4.34·17-s + 4.75·19-s − 0.241·21-s + 3.16·23-s + 6.49·25-s + 1.74·27-s + 2.75·29-s + 0.312·31-s + 0.323·33-s − 2.77·35-s − 7.02·37-s + 0.320·39-s + 3.34·41-s + 8.25·43-s + 9.87·45-s − 0.104·47-s − 6.33·49-s − 1.28·51-s − 3.94·53-s + 3.71·55-s + ⋯ |
| L(s) = 1 | − 0.170·3-s − 1.51·5-s + 0.309·7-s − 0.970·9-s − 0.330·11-s − 0.301·13-s + 0.258·15-s + 1.05·17-s + 1.09·19-s − 0.0526·21-s + 0.659·23-s + 1.29·25-s + 0.335·27-s + 0.511·29-s + 0.0560·31-s + 0.0562·33-s − 0.468·35-s − 1.15·37-s + 0.0513·39-s + 0.522·41-s + 1.25·43-s + 1.47·45-s − 0.0152·47-s − 0.904·49-s − 0.179·51-s − 0.542·53-s + 0.500·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 0.294T + 3T^{2} \) |
| 5 | \( 1 + 3.39T + 5T^{2} \) |
| 7 | \( 1 - 0.817T + 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 + 1.08T + 13T^{2} \) |
| 17 | \( 1 - 4.34T + 17T^{2} \) |
| 19 | \( 1 - 4.75T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 - 2.75T + 29T^{2} \) |
| 31 | \( 1 - 0.312T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 - 3.34T + 41T^{2} \) |
| 43 | \( 1 - 8.25T + 43T^{2} \) |
| 47 | \( 1 + 0.104T + 47T^{2} \) |
| 53 | \( 1 + 3.94T + 53T^{2} \) |
| 59 | \( 1 - 2.50T + 59T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 9.42T + 71T^{2} \) |
| 73 | \( 1 + 1.26T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 7.23T + 89T^{2} \) |
| 97 | \( 1 + 5.07T + 97T^{2} \) |
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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
−7.920032865040992719510483207518, −7.58596420520373671607717084271, −6.77999477012352517823654386181, −5.66546635112136242165937627083, −5.10920643097520626875280277379, −4.27184499537302090146500618881, −3.31019256532059595747955686555, −2.81328076116587888130456133165, −1.15120338923937659355245831759, 0,
1.15120338923937659355245831759, 2.81328076116587888130456133165, 3.31019256532059595747955686555, 4.27184499537302090146500618881, 5.10920643097520626875280277379, 5.66546635112136242165937627083, 6.77999477012352517823654386181, 7.58596420520373671607717084271, 7.920032865040992719510483207518
