| L(s) = 1 | + 1.22·2-s + 0.651·3-s − 0.489·4-s + 0.800·6-s + 3.71·7-s − 3.05·8-s − 2.57·9-s + 0.806·11-s − 0.319·12-s + 1.77·13-s + 4.56·14-s − 2.78·16-s + 6.55·17-s − 3.16·18-s + 4.55·19-s + 2.42·21-s + 0.990·22-s + 4.91·23-s − 1.99·24-s + 2.18·26-s − 3.63·27-s − 1.82·28-s − 0.0967·29-s + 1.58·31-s + 2.70·32-s + 0.525·33-s + 8.05·34-s + ⋯ |
| L(s) = 1 | + 0.868·2-s + 0.376·3-s − 0.244·4-s + 0.326·6-s + 1.40·7-s − 1.08·8-s − 0.858·9-s + 0.243·11-s − 0.0921·12-s + 0.492·13-s + 1.22·14-s − 0.695·16-s + 1.59·17-s − 0.746·18-s + 1.04·19-s + 0.528·21-s + 0.211·22-s + 1.02·23-s − 0.406·24-s + 0.428·26-s − 0.699·27-s − 0.344·28-s − 0.0179·29-s + 0.284·31-s + 0.477·32-s + 0.0914·33-s + 1.38·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.819095334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.819095334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 3 | \( 1 - 0.651T + 3T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 - 0.806T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 - 6.55T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 + 0.0967T + 29T^{2} \) |
| 31 | \( 1 - 1.58T + 31T^{2} \) |
| 37 | \( 1 - 1.74T + 37T^{2} \) |
| 41 | \( 1 - 0.954T + 41T^{2} \) |
| 47 | \( 1 + 4.00T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 1.94T + 59T^{2} \) |
| 61 | \( 1 + 8.31T + 61T^{2} \) |
| 67 | \( 1 - 1.31T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 9.54T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 1.06T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−9.730173554386816193991985202440, −8.973202627478447041345643867169, −8.205252641083906484769455553861, −7.59072379883653687977364838000, −6.17281217343388985003862352007, −5.35061871386470486268244354928, −4.80796276131034012374376209530, −3.61327971543777199102649003475, −2.89538789129409458860185624103, −1.26489163293096752437328677361,
1.26489163293096752437328677361, 2.89538789129409458860185624103, 3.61327971543777199102649003475, 4.80796276131034012374376209530, 5.35061871386470486268244354928, 6.17281217343388985003862352007, 7.59072379883653687977364838000, 8.205252641083906484769455553861, 8.973202627478447041345643867169, 9.730173554386816193991985202440
