| L(s) = 1 | − 2.39·2-s + 3.71·4-s − 3.15·5-s + 2.42·7-s − 4.09·8-s + 7.55·10-s + 4.78·11-s − 2.17·13-s − 5.80·14-s + 2.36·16-s + 2.29·17-s + 0.938·19-s − 11.7·20-s − 11.4·22-s + 23-s + 4.97·25-s + 5.20·26-s + 9.02·28-s + 29-s + 2.03·31-s + 2.53·32-s − 5.48·34-s − 7.67·35-s + 8.81·37-s − 2.24·38-s + 12.9·40-s + 4.46·41-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.85·4-s − 1.41·5-s + 0.918·7-s − 1.44·8-s + 2.38·10-s + 1.44·11-s − 0.603·13-s − 1.55·14-s + 0.591·16-s + 0.556·17-s + 0.215·19-s − 2.62·20-s − 2.43·22-s + 0.208·23-s + 0.995·25-s + 1.02·26-s + 1.70·28-s + 0.185·29-s + 0.364·31-s + 0.448·32-s − 0.940·34-s − 1.29·35-s + 1.44·37-s − 0.363·38-s + 2.04·40-s + 0.698·41-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
| \(L(1)\) |
\(\approx\) |
\(0.7800334019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7800334019\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 - 4.78T + 11T^{2} \) |
| 13 | \( 1 + 2.17T + 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 19 | \( 1 - 0.938T + 19T^{2} \) |
| 31 | \( 1 - 2.03T + 31T^{2} \) |
| 37 | \( 1 - 8.81T + 37T^{2} \) |
| 41 | \( 1 - 4.46T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 8.57T + 47T^{2} \) |
| 53 | \( 1 - 9.87T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 6.87T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 0.894T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 1.79T + 79T^{2} \) |
| 83 | \( 1 - 1.59T + 83T^{2} \) |
| 89 | \( 1 - 5.46T + 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
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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
−8.021715619873106535213011955257, −7.76775335533696829918675969664, −6.95466250168247651411165840740, −6.46417093287334875894922425343, −5.16647550822506490941550203025, −4.32710735410906540556707184485, −3.59546104163950762964048532045, −2.47325955455126552048172715323, −1.38675479236062530084153454472, −0.67204369383267270694003129666,
0.67204369383267270694003129666, 1.38675479236062530084153454472, 2.47325955455126552048172715323, 3.59546104163950762964048532045, 4.32710735410906540556707184485, 5.16647550822506490941550203025, 6.46417093287334875894922425343, 6.95466250168247651411165840740, 7.76775335533696829918675969664, 8.021715619873106535213011955257
