| L(s) = 1 | − 2.28·2-s + 2.50·3-s + 3.22·4-s − 5.72·6-s − 3.50·7-s − 2.79·8-s + 3.28·9-s − 4.50·11-s + 8.07·12-s + 5·13-s + 8.01·14-s − 0.0632·16-s − 0.158·17-s − 7.50·18-s − 8.79·21-s + 10.2·22-s − 1.15·23-s − 7.00·24-s − 11.4·26-s + 0.714·27-s − 11.2·28-s − 3.50·29-s − 2.28·31-s + 5.72·32-s − 11.2·33-s + 0.362·34-s + 10.5·36-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.44·3-s + 1.61·4-s − 2.33·6-s − 1.32·7-s − 0.987·8-s + 1.09·9-s − 1.35·11-s + 2.33·12-s + 1.38·13-s + 2.14·14-s − 0.0158·16-s − 0.0384·17-s − 1.76·18-s − 1.91·21-s + 2.19·22-s − 0.241·23-s − 1.42·24-s − 2.24·26-s + 0.137·27-s − 2.13·28-s − 0.651·29-s − 0.410·31-s + 1.01·32-s − 1.96·33-s + 0.0621·34-s + 1.76·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 - 2.50T + 3T^{2} \) |
| 7 | \( 1 + 3.50T + 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + 0.158T + 17T^{2} \) |
| 23 | \( 1 + 1.15T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6.07T + 41T^{2} \) |
| 43 | \( 1 - 3.34T + 43T^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 - 5.74T + 53T^{2} \) |
| 59 | \( 1 - 3.06T + 59T^{2} \) |
| 61 | \( 1 + 0.873T + 61T^{2} \) |
| 67 | \( 1 - 8.44T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 - 7.15T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 4.85T + 83T^{2} \) |
| 89 | \( 1 + 1.11T + 89T^{2} \) |
| 97 | \( 1 + 1.61T + 97T^{2} \) |
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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
−7.72793966703857865166004989771, −7.14438720303147320593363023470, −6.28339607799666082641472305939, −5.68416946619165351927972367196, −4.24451524364556027227331857266, −3.48815395761093242644214061674, −2.73629143767410517671669607683, −2.26531702786880219725798819481, −1.14069001456192367438933525297, 0,
1.14069001456192367438933525297, 2.26531702786880219725798819481, 2.73629143767410517671669607683, 3.48815395761093242644214061674, 4.24451524364556027227331857266, 5.68416946619165351927972367196, 6.28339607799666082641472305939, 7.14438720303147320593363023470, 7.72793966703857865166004989771
