| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 6·11-s − 4·13-s − 14-s + 16-s + 6·17-s − 4·19-s − 20-s − 6·22-s + 23-s + 25-s + 4·26-s + 28-s − 4·31-s − 32-s − 6·34-s − 35-s − 10·37-s + 4·38-s + 40-s − 6·41-s − 10·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.80·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.223·20-s − 1.27·22-s + 0.208·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s − 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.169·35-s − 1.64·37-s + 0.648·38-s + 0.158·40-s − 0.937·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.413908950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.413908950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.46506914366329, −15.42341674541012, −15.01946068593611, −14.47737590458674, −14.21071612090927, −13.28334428982766, −12.34137525584646, −12.09882772937403, −11.70003179823788, −10.97321763701567, −10.32770424758076, −9.781312041557178, −9.207714266256477, −8.579817998759231, −8.107125472020328, −7.329524622598250, −6.868803343428496, −6.325264449795465, −5.317654474250589, −4.815743354489274, −3.691422701693929, −3.485516292586876, −2.189706167996309, −1.556669872608677, −0.6029107506558618,
0.6029107506558618, 1.556669872608677, 2.189706167996309, 3.485516292586876, 3.691422701693929, 4.815743354489274, 5.317654474250589, 6.325264449795465, 6.868803343428496, 7.329524622598250, 8.107125472020328, 8.579817998759231, 9.207714266256477, 9.781312041557178, 10.32770424758076, 10.97321763701567, 11.70003179823788, 12.09882772937403, 12.34137525584646, 13.28334428982766, 14.21071612090927, 14.47737590458674, 15.01946068593611, 15.42341674541012, 16.46506914366329
