| L(s) = 1 | + 3-s − 2·5-s − 2·7-s + 9-s + 11-s + 2·13-s − 2·15-s − 4·17-s + 6·19-s − 2·21-s − 25-s + 27-s + 8·29-s + 8·31-s + 33-s + 4·35-s + 2·39-s + 8·41-s + 2·43-s − 2·45-s − 8·47-s − 3·49-s − 4·51-s − 2·53-s − 2·55-s + 6·57-s − 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.516·15-s − 0.970·17-s + 1.37·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s + 1.48·29-s + 1.43·31-s + 0.174·33-s + 0.676·35-s + 0.320·39-s + 1.24·41-s + 0.304·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s − 0.269·55-s + 0.794·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180708 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180708 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.620186940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.620186940\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.31115682934739, −12.57957401866187, −12.24727126022240, −11.84275934026734, −11.24178741752836, −10.91305563185886, −10.27893196005356, −9.658110854451455, −9.418206653426974, −8.890469824006272, −8.298019714843856, −7.853274707838722, −7.603905828981407, −6.766053392134232, −6.404431389449822, −6.137385003701118, −5.048253162821801, −4.764870476784664, −4.026278836137061, −3.702192744802590, −3.001214197942216, −2.773056814308617, −1.874402231159305, −1.061388585700920, −0.5122658817417255,
0.5122658817417255, 1.061388585700920, 1.874402231159305, 2.773056814308617, 3.001214197942216, 3.702192744802590, 4.026278836137061, 4.764870476784664, 5.048253162821801, 6.137385003701118, 6.404431389449822, 6.766053392134232, 7.603905828981407, 7.853274707838722, 8.298019714843856, 8.890469824006272, 9.418206653426974, 9.658110854451455, 10.27893196005356, 10.91305563185886, 11.24178741752836, 11.84275934026734, 12.24727126022240, 12.57957401866187, 13.31115682934739
