| L(s) = 1 | − 2.86·3-s − 0.462·5-s − 7-s + 5.18·9-s − 2·11-s + 6.64·13-s + 1.32·15-s − 17-s + 4.64·19-s + 2.86·21-s − 5.72·23-s − 4.78·25-s − 6.24·27-s + 0.796·29-s + 0.184·31-s + 5.72·33-s + 0.462·35-s − 7.44·37-s − 19.0·39-s − 6.71·41-s + 5.50·43-s − 2.39·45-s − 5.72·47-s + 49-s + 2.86·51-s − 4.33·53-s + 0.925·55-s + ⋯ |
| L(s) = 1 | − 1.65·3-s − 0.206·5-s − 0.377·7-s + 1.72·9-s − 0.603·11-s + 1.84·13-s + 0.341·15-s − 0.242·17-s + 1.06·19-s + 0.624·21-s − 1.19·23-s − 0.957·25-s − 1.20·27-s + 0.147·29-s + 0.0330·31-s + 0.996·33-s + 0.0781·35-s − 1.22·37-s − 3.04·39-s − 1.04·41-s + 0.839·43-s − 0.357·45-s − 0.834·47-s + 0.142·49-s + 0.400·51-s − 0.595·53-s + 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.86T + 3T^{2} \) |
| 5 | \( 1 + 0.462T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 6.64T + 13T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 - 0.796T + 29T^{2} \) |
| 31 | \( 1 - 0.184T + 31T^{2} \) |
| 37 | \( 1 + 7.44T + 37T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 - 5.50T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 + 4.33T + 53T^{2} \) |
| 59 | \( 1 - 0.646T + 59T^{2} \) |
| 61 | \( 1 - 4.86T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 0.556T + 71T^{2} \) |
| 73 | \( 1 + 4.73T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 6.36T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−10.01963709612641236726483165130, −8.795303434198117177399428466590, −7.80788700663037247956075969132, −6.82448387374734144699018174841, −5.98958117270713965028109659456, −5.53605431312127747713034786267, −4.39107868653115196038367174544, −3.40290271568107814091269375596, −1.46883846965923117130653728435, 0,
1.46883846965923117130653728435, 3.40290271568107814091269375596, 4.39107868653115196038367174544, 5.53605431312127747713034786267, 5.98958117270713965028109659456, 6.82448387374734144699018174841, 7.80788700663037247956075969132, 8.795303434198117177399428466590, 10.01963709612641236726483165130
