| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 4·7-s − 8-s + 9-s + 10-s + 6·11-s − 12-s + 2·13-s − 4·14-s + 15-s + 16-s − 18-s + 19-s − 20-s − 4·21-s − 6·22-s + 24-s + 25-s − 2·26-s − 27-s + 4·28-s − 30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s + 0.554·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.872·21-s − 1.27·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s − 0.182·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664300410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664300410\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.31162331298646, −12.42289666531507, −12.11237774148280, −11.70804191341330, −11.32203550137289, −10.99624862700790, −10.55729197910659, −9.914935881965374, −9.329757636268839, −8.878822441882386, −8.490198502901227, −7.981181963125993, −7.502460132988373, −6.826843824550909, −6.655719772855573, −5.957094559448062, −5.271710786980410, −4.925726308327458, −4.241260001956312, −3.655834109764550, −3.318682061227677, −2.092068990484165, −1.562552179838610, −1.334409675182624, −0.4596651311157840,
0.4596651311157840, 1.334409675182624, 1.562552179838610, 2.092068990484165, 3.318682061227677, 3.655834109764550, 4.241260001956312, 4.925726308327458, 5.271710786980410, 5.957094559448062, 6.655719772855573, 6.826843824550909, 7.502460132988373, 7.981181963125993, 8.490198502901227, 8.878822441882386, 9.329757636268839, 9.914935881965374, 10.55729197910659, 10.99624862700790, 11.32203550137289, 11.70804191341330, 12.11237774148280, 12.42289666531507, 13.31162331298646
