| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 7-s − 8-s + 9-s − 2·10-s − 4·11-s − 12-s − 6·13-s + 14-s − 2·15-s + 16-s − 2·17-s − 18-s + 4·19-s + 2·20-s + 21-s + 4·22-s − 8·23-s + 24-s − 25-s + 6·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.64122237504473, −13.92433733254927, −13.71809604555050, −13.01400336703566, −12.43920256788808, −12.10118858525421, −11.57931901331636, −10.88354391399767, −10.27440147924864, −10.03616574640424, −9.641015355086366, −9.176140614799169, −8.308082593510263, −7.809132616016500, −7.320568134055959, −6.806445301454516, −6.082907373405239, −5.717769197377252, −5.090500520263713, −4.647781074565769, −3.673633074964561, −2.840671428801989, −2.227705012320293, −1.880046146227356, −0.6829364616712145, 0,
0.6829364616712145, 1.880046146227356, 2.227705012320293, 2.840671428801989, 3.673633074964561, 4.647781074565769, 5.090500520263713, 5.717769197377252, 6.082907373405239, 6.806445301454516, 7.320568134055959, 7.809132616016500, 8.308082593510263, 9.176140614799169, 9.641015355086366, 10.03616574640424, 10.27440147924864, 10.88354391399767, 11.57931901331636, 12.10118858525421, 12.43920256788808, 13.01400336703566, 13.71809604555050, 13.92433733254927, 14.64122237504473
