| L(s) = 1 | + 0.289·2-s − 1.91·4-s − 0.894·5-s + 3.31·7-s − 1.13·8-s − 0.258·10-s − 3.47·11-s + 1.33·13-s + 0.957·14-s + 3.50·16-s + 4.10·17-s − 2.37·19-s + 1.71·20-s − 1.00·22-s + 23-s − 4.20·25-s + 0.385·26-s − 6.34·28-s + 29-s − 8.16·31-s + 3.27·32-s + 1.18·34-s − 2.96·35-s − 7.23·37-s − 0.686·38-s + 1.01·40-s + 2.74·41-s + ⋯ |
| L(s) = 1 | + 0.204·2-s − 0.958·4-s − 0.399·5-s + 1.25·7-s − 0.400·8-s − 0.0817·10-s − 1.04·11-s + 0.370·13-s + 0.255·14-s + 0.876·16-s + 0.995·17-s − 0.544·19-s + 0.383·20-s − 0.214·22-s + 0.208·23-s − 0.840·25-s + 0.0756·26-s − 1.19·28-s + 0.185·29-s − 1.46·31-s + 0.579·32-s + 0.203·34-s − 0.500·35-s − 1.18·37-s − 0.111·38-s + 0.160·40-s + 0.428·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.289T + 2T^{2} \) |
| 5 | \( 1 + 0.894T + 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 - 4.10T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 31 | \( 1 + 8.16T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 - 1.38T + 43T^{2} \) |
| 47 | \( 1 - 2.36T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 15.2T + 59T^{2} \) |
| 61 | \( 1 - 4.80T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 0.717T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 - 8.33T + 79T^{2} \) |
| 83 | \( 1 + 0.376T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.73402600994961633808686349435, −7.37493936639894499955252505862, −6.03503521158660557078169441642, −5.33575450129885613346788753804, −4.96206401075285416280076106091, −4.03799857025117016791334914952, −3.50389248625558640976919419362, −2.32326871620683720800583674605, −1.24424332575653202718190008361, 0,
1.24424332575653202718190008361, 2.32326871620683720800583674605, 3.50389248625558640976919419362, 4.03799857025117016791334914952, 4.96206401075285416280076106091, 5.33575450129885613346788753804, 6.03503521158660557078169441642, 7.37493936639894499955252505862, 7.73402600994961633808686349435
