| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 12-s + 2·13-s + 14-s + 15-s + 16-s − 2·17-s + 18-s − 20-s − 21-s + 8·23-s − 24-s + 25-s + 2·26-s − 27-s + 28-s − 2·29-s + 30-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.223·20-s − 0.218·21-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.182·30-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.915941637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.915941637\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.31316141884498, −14.89331423734258, −14.30402214453833, −13.62421768130874, −13.24589391102570, −12.51779645813329, −12.30074425005263, −11.44882674690523, −11.01553778720336, −10.92771200848440, −10.01512338953745, −9.342694472103051, −8.622048686724738, −8.134985503510658, −7.337107102587300, −6.836026823148038, −6.405545253564514, −5.494764792976137, −5.130286966384583, −4.519765101521246, −3.805938498424054, −3.267644237908575, −2.353720375450516, −1.503750063741943, −0.6459249915511187,
0.6459249915511187, 1.503750063741943, 2.353720375450516, 3.267644237908575, 3.805938498424054, 4.519765101521246, 5.130286966384583, 5.494764792976137, 6.405545253564514, 6.836026823148038, 7.337107102587300, 8.134985503510658, 8.622048686724738, 9.342694472103051, 10.01512338953745, 10.92771200848440, 11.01553778720336, 11.44882674690523, 12.30074425005263, 12.51779645813329, 13.24589391102570, 13.62421768130874, 14.30402214453833, 14.89331423734258, 15.31316141884498
