| L(s) = 1 | − 0.851·2-s − 3-s − 1.27·4-s + 0.851·6-s + 0.870·7-s + 2.78·8-s + 9-s + 2.29·11-s + 1.27·12-s − 3.00·13-s − 0.741·14-s + 0.173·16-s − 5.87·17-s − 0.851·18-s − 0.0882·19-s − 0.870·21-s − 1.95·22-s + 6.70·23-s − 2.78·24-s + 2.56·26-s − 27-s − 1.10·28-s − 1.87·29-s + 5.65·31-s − 5.72·32-s − 2.29·33-s + 5.00·34-s + ⋯ |
| L(s) = 1 | − 0.602·2-s − 0.577·3-s − 0.637·4-s + 0.347·6-s + 0.329·7-s + 0.986·8-s + 0.333·9-s + 0.690·11-s + 0.367·12-s − 0.833·13-s − 0.198·14-s + 0.0432·16-s − 1.42·17-s − 0.200·18-s − 0.0202·19-s − 0.190·21-s − 0.415·22-s + 1.39·23-s − 0.569·24-s + 0.502·26-s − 0.192·27-s − 0.209·28-s − 0.348·29-s + 1.01·31-s − 1.01·32-s − 0.398·33-s + 0.858·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 + 0.851T + 2T^{2} \) |
| 7 | \( 1 - 0.870T + 7T^{2} \) |
| 11 | \( 1 - 2.29T + 11T^{2} \) |
| 13 | \( 1 + 3.00T + 13T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 + 0.0882T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 3.52T + 37T^{2} \) |
| 41 | \( 1 - 5.59T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 + 9.64T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 5.22T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 3.01T + 71T^{2} \) |
| 73 | \( 1 + 8.03T + 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 - 6.63T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.56710428907481531847334596717, −6.81457502094899432417827983762, −6.27678177088911048140239117820, −5.19368337100850188618286857107, −4.67518675470425018146815341510, −4.22559786896916942625396907808, −3.07255314862966178581381505593, −1.93683638627865819822144401731, −1.03471672740158607358909123882, 0,
1.03471672740158607358909123882, 1.93683638627865819822144401731, 3.07255314862966178581381505593, 4.22559786896916942625396907808, 4.67518675470425018146815341510, 5.19368337100850188618286857107, 6.27678177088911048140239117820, 6.81457502094899432417827983762, 7.56710428907481531847334596717
