| L(s) = 1 | + 0.445·2-s − 1.80·4-s − 5-s + 0.554·7-s − 1.69·8-s − 0.445·10-s + 11-s + 0.246·14-s + 2.85·16-s + 3.10·17-s − 6.40·19-s + 1.80·20-s + 0.445·22-s − 3.24·23-s + 25-s − 28-s − 4.51·29-s + 7.09·31-s + 4.65·32-s + 1.38·34-s − 0.554·35-s + 3.58·37-s − 2.85·38-s + 1.69·40-s + 7.44·41-s − 5.44·43-s − 1.80·44-s + ⋯ |
| L(s) = 1 | + 0.314·2-s − 0.900·4-s − 0.447·5-s + 0.209·7-s − 0.598·8-s − 0.140·10-s + 0.301·11-s + 0.0660·14-s + 0.712·16-s + 0.754·17-s − 1.46·19-s + 0.402·20-s + 0.0948·22-s − 0.677·23-s + 0.200·25-s − 0.188·28-s − 0.838·29-s + 1.27·31-s + 0.822·32-s + 0.237·34-s − 0.0938·35-s + 0.588·37-s − 0.462·38-s + 0.267·40-s + 1.16·41-s − 0.830·43-s − 0.271·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.445T + 2T^{2} \) |
| 7 | \( 1 - 0.554T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 + 6.40T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 + 4.51T + 29T^{2} \) |
| 31 | \( 1 - 7.09T + 31T^{2} \) |
| 37 | \( 1 - 3.58T + 37T^{2} \) |
| 41 | \( 1 - 7.44T + 41T^{2} \) |
| 43 | \( 1 + 5.44T + 43T^{2} \) |
| 47 | \( 1 - 1.76T + 47T^{2} \) |
| 53 | \( 1 - 7.92T + 53T^{2} \) |
| 59 | \( 1 + 4.71T + 59T^{2} \) |
| 61 | \( 1 - 7.72T + 61T^{2} \) |
| 67 | \( 1 + 2.14T + 67T^{2} \) |
| 71 | \( 1 + 7.07T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 1.48T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 9.39T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−7.74894651818745929699376176691, −6.73515179487412845911674128341, −6.04012582776589752939459321286, −5.38592768009525738793301072814, −4.48236844812904987369286748736, −4.11145662116574323589148909478, −3.34459966017591861021475588137, −2.35336988950309656285346866061, −1.12673523941713530237825658554, 0,
1.12673523941713530237825658554, 2.35336988950309656285346866061, 3.34459966017591861021475588137, 4.11145662116574323589148909478, 4.48236844812904987369286748736, 5.38592768009525738793301072814, 6.04012582776589752939459321286, 6.73515179487412845911674128341, 7.74894651818745929699376176691
