| L(s) = 1 | − 0.669·2-s − 1.55·4-s − 2.14·5-s + 7-s + 2.37·8-s + 1.43·10-s + 2.52·13-s − 0.669·14-s + 1.50·16-s + 1.79·17-s − 6.72·19-s + 3.32·20-s + 3.16·23-s − 0.404·25-s − 1.69·26-s − 1.55·28-s + 0.924·29-s + 3.00·31-s − 5.76·32-s − 1.19·34-s − 2.14·35-s − 1.50·37-s + 4.50·38-s − 5.09·40-s − 5.56·41-s − 8.42·43-s − 2.11·46-s + ⋯ |
| L(s) = 1 | − 0.473·2-s − 0.775·4-s − 0.958·5-s + 0.377·7-s + 0.840·8-s + 0.454·10-s + 0.701·13-s − 0.178·14-s + 0.377·16-s + 0.434·17-s − 1.54·19-s + 0.743·20-s + 0.659·23-s − 0.0808·25-s − 0.332·26-s − 0.293·28-s + 0.171·29-s + 0.539·31-s − 1.01·32-s − 0.205·34-s − 0.362·35-s − 0.247·37-s + 0.730·38-s − 0.806·40-s − 0.868·41-s − 1.28·43-s − 0.312·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.669T + 2T^{2} \) |
| 5 | \( 1 + 2.14T + 5T^{2} \) |
| 13 | \( 1 - 2.52T + 13T^{2} \) |
| 17 | \( 1 - 1.79T + 17T^{2} \) |
| 19 | \( 1 + 6.72T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 - 0.924T + 29T^{2} \) |
| 31 | \( 1 - 3.00T + 31T^{2} \) |
| 37 | \( 1 + 1.50T + 37T^{2} \) |
| 41 | \( 1 + 5.56T + 41T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 + 4.39T + 47T^{2} \) |
| 53 | \( 1 + 0.667T + 53T^{2} \) |
| 59 | \( 1 - 0.368T + 59T^{2} \) |
| 61 | \( 1 - 5.01T + 61T^{2} \) |
| 67 | \( 1 + 0.902T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 8.03T + 73T^{2} \) |
| 79 | \( 1 - 4.05T + 79T^{2} \) |
| 83 | \( 1 - 4.05T + 83T^{2} \) |
| 89 | \( 1 - 8.30T + 89T^{2} \) |
| 97 | \( 1 - 8.51T + 97T^{2} \) |
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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
−7.84010689994515108855630335645, −6.94440030328719393540001347737, −6.27858931971005371973686986573, −5.19254962837561543726553012951, −4.69372230113894755514765800926, −3.88089933079657475135014760430, −3.41098313774011483098330192541, −2.04678160554319797401908399048, −1.02370960388510578666082169785, 0,
1.02370960388510578666082169785, 2.04678160554319797401908399048, 3.41098313774011483098330192541, 3.88089933079657475135014760430, 4.69372230113894755514765800926, 5.19254962837561543726553012951, 6.27858931971005371973686986573, 6.94440030328719393540001347737, 7.84010689994515108855630335645
