| L(s) = 1 | + 0.162·2-s − 1.97·4-s + 4.38·5-s + 7-s − 0.644·8-s + 0.711·10-s − 1.67·13-s + 0.162·14-s + 3.84·16-s + 6.18·17-s + 3.47·19-s − 8.65·20-s − 0.568·23-s + 14.2·25-s − 0.271·26-s − 1.97·28-s − 8.86·29-s + 4.33·31-s + 1.91·32-s + 1.00·34-s + 4.38·35-s + 0.969·37-s + 0.563·38-s − 2.82·40-s + 5.77·41-s + 5.04·43-s − 0.0921·46-s + ⋯ |
| L(s) = 1 | + 0.114·2-s − 0.986·4-s + 1.96·5-s + 0.377·7-s − 0.227·8-s + 0.224·10-s − 0.464·13-s + 0.0433·14-s + 0.960·16-s + 1.50·17-s + 0.797·19-s − 1.93·20-s − 0.118·23-s + 2.84·25-s − 0.0532·26-s − 0.372·28-s − 1.64·29-s + 0.778·31-s + 0.337·32-s + 0.172·34-s + 0.741·35-s + 0.159·37-s + 0.0914·38-s − 0.446·40-s + 0.902·41-s + 0.769·43-s − 0.0135·46-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
| \(L(1)\) |
\(\approx\) |
\(2.962168400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.962168400\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.162T + 2T^{2} \) |
| 5 | \( 1 - 4.38T + 5T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 - 6.18T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 23 | \( 1 + 0.568T + 23T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 - 4.33T + 31T^{2} \) |
| 37 | \( 1 - 0.969T + 37T^{2} \) |
| 41 | \( 1 - 5.77T + 41T^{2} \) |
| 43 | \( 1 - 5.04T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 - 2.37T + 53T^{2} \) |
| 59 | \( 1 - 2.84T + 59T^{2} \) |
| 61 | \( 1 + 9.29T + 61T^{2} \) |
| 67 | \( 1 + 7.14T + 67T^{2} \) |
| 71 | \( 1 - 0.794T + 71T^{2} \) |
| 73 | \( 1 + 2.38T + 73T^{2} \) |
| 79 | \( 1 - 0.670T + 79T^{2} \) |
| 83 | \( 1 + 9.28T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.83791618651444243586511459499, −7.28853093993346242502165661635, −6.10627552455838696927702989247, −5.72976689926332702283622954120, −5.19200752088708261241976640210, −4.56239202243674935957118415622, −3.46184207845872961317683322444, −2.68089949213873496284019526136, −1.68504486501141907441597385027, −0.921675395505286575241811086068,
0.921675395505286575241811086068, 1.68504486501141907441597385027, 2.68089949213873496284019526136, 3.46184207845872961317683322444, 4.56239202243674935957118415622, 5.19200752088708261241976640210, 5.72976689926332702283622954120, 6.10627552455838696927702989247, 7.28853093993346242502165661635, 7.83791618651444243586511459499
