| L(s) = 1 | + 2·2-s + 4·4-s + 21·5-s + 7·7-s + 8·8-s + 42·10-s + 11·11-s + 65·13-s + 14·14-s + 16·16-s + 54·17-s + 65·19-s + 84·20-s + 22·22-s − 132·23-s + 316·25-s + 130·26-s + 28·28-s − 39·29-s − 178·31-s + 32·32-s + 108·34-s + 147·35-s − 439·37-s + 130·38-s + 168·40-s − 96·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.87·5-s + 0.377·7-s + 0.353·8-s + 1.32·10-s + 0.301·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.770·17-s + 0.784·19-s + 0.939·20-s + 0.213·22-s − 1.19·23-s + 2.52·25-s + 0.980·26-s + 0.188·28-s − 0.249·29-s − 1.03·31-s + 0.176·32-s + 0.544·34-s + 0.709·35-s − 1.95·37-s + 0.554·38-s + 0.664·40-s − 0.365·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.144854769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.144854769\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 5 | \( 1 - 21 T + p^{3} T^{2} \) |
| 13 | \( 1 - 5 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 65 T + p^{3} T^{2} \) |
| 23 | \( 1 + 132 T + p^{3} T^{2} \) |
| 29 | \( 1 + 39 T + p^{3} T^{2} \) |
| 31 | \( 1 + 178 T + p^{3} T^{2} \) |
| 37 | \( 1 + 439 T + p^{3} T^{2} \) |
| 41 | \( 1 + 96 T + p^{3} T^{2} \) |
| 43 | \( 1 - 272 T + p^{3} T^{2} \) |
| 47 | \( 1 - 375 T + p^{3} T^{2} \) |
| 53 | \( 1 + 612 T + p^{3} T^{2} \) |
| 59 | \( 1 - 507 T + p^{3} T^{2} \) |
| 61 | \( 1 - 758 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1087 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 673 T + p^{3} T^{2} \) |
| 79 | \( 1 + 700 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1218 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1350 T + p^{3} T^{2} \) |
| 97 | \( 1 + 808 T + p^{3} T^{2} \) |
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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
−9.251315113627616476765556324442, −8.544772424362594359208804706336, −7.36756350756545770223643597489, −6.45973340222178331064372305974, −5.63830088102197312161318375808, −5.43260277518394766263374598608, −4.06119353596282885218467138213, −3.08460171102225907013749713392, −1.88066958820505151717300657094, −1.30134538464155866186832246875,
1.30134538464155866186832246875, 1.88066958820505151717300657094, 3.08460171102225907013749713392, 4.06119353596282885218467138213, 5.43260277518394766263374598608, 5.63830088102197312161318375808, 6.45973340222178331064372305974, 7.36756350756545770223643597489, 8.544772424362594359208804706336, 9.251315113627616476765556324442
