| L(s) = 1 | − 2-s + 1.58·3-s + 4-s + 4.43·5-s − 1.58·6-s + 2.05·7-s − 8-s − 0.477·9-s − 4.43·10-s − 2.29·11-s + 1.58·12-s + 1.35·13-s − 2.05·14-s + 7.05·15-s + 16-s − 5.61·17-s + 0.477·18-s − 19-s + 4.43·20-s + 3.26·21-s + 2.29·22-s + 6.75·23-s − 1.58·24-s + 14.7·25-s − 1.35·26-s − 5.52·27-s + 2.05·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.916·3-s + 0.5·4-s + 1.98·5-s − 0.648·6-s + 0.776·7-s − 0.353·8-s − 0.159·9-s − 1.40·10-s − 0.693·11-s + 0.458·12-s + 0.376·13-s − 0.549·14-s + 1.82·15-s + 0.250·16-s − 1.36·17-s + 0.112·18-s − 0.229·19-s + 0.992·20-s + 0.712·21-s + 0.490·22-s + 1.40·23-s − 0.324·24-s + 2.94·25-s − 0.266·26-s − 1.06·27-s + 0.388·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.363429720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.363429720\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 - 4.43T + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 5.61T + 17T^{2} \) |
| 23 | \( 1 - 6.75T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 + 4.55T + 31T^{2} \) |
| 37 | \( 1 - 2.78T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 - 1.84T + 43T^{2} \) |
| 47 | \( 1 - 9.53T + 47T^{2} \) |
| 53 | \( 1 - 4.52T + 53T^{2} \) |
| 59 | \( 1 - 5.82T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 - 0.756T + 67T^{2} \) |
| 71 | \( 1 - 6.75T + 71T^{2} \) |
| 73 | \( 1 - 4.44T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 + 6.04T + 89T^{2} \) |
| 97 | \( 1 - 2.42T + 97T^{2} \) |
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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
−8.123178010276032152224487762271, −7.16565484040967828683840235911, −6.58025489317412602292841189334, −5.75247043213151236941616130048, −5.25539839293848827967892829696, −4.34787029650450671602951100062, −2.98065780971069236825611089728, −2.44013408327530144820684506355, −1.96296676729610004102379058068, −0.993983092055731368048445272366,
0.993983092055731368048445272366, 1.96296676729610004102379058068, 2.44013408327530144820684506355, 2.98065780971069236825611089728, 4.34787029650450671602951100062, 5.25539839293848827967892829696, 5.75247043213151236941616130048, 6.58025489317412602292841189334, 7.16565484040967828683840235911, 8.123178010276032152224487762271
