| L(s) = 1 | − 0.0794·2-s + 0.864·3-s − 1.99·4-s − 0.0687·6-s − 0.264·7-s + 0.317·8-s − 2.25·9-s + 3.42·11-s − 1.72·12-s − 2.24·13-s + 0.0209·14-s + 3.96·16-s + 1.58·17-s + 0.179·18-s − 7.48·19-s − 0.228·21-s − 0.272·22-s + 0.593·23-s + 0.274·24-s + 0.178·26-s − 4.54·27-s + 0.526·28-s + 6.21·29-s − 5.73·31-s − 0.949·32-s + 2.96·33-s − 0.126·34-s + ⋯ |
| L(s) = 1 | − 0.0562·2-s + 0.499·3-s − 0.996·4-s − 0.0280·6-s − 0.0997·7-s + 0.112·8-s − 0.750·9-s + 1.03·11-s − 0.497·12-s − 0.622·13-s + 0.00560·14-s + 0.990·16-s + 0.384·17-s + 0.0421·18-s − 1.71·19-s − 0.0498·21-s − 0.0580·22-s + 0.123·23-s + 0.0560·24-s + 0.0349·26-s − 0.874·27-s + 0.0994·28-s + 1.15·29-s − 1.02·31-s − 0.167·32-s + 0.515·33-s − 0.0216·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.308998171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308998171\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0794T + 2T^{2} \) |
| 3 | \( 1 - 0.864T + 3T^{2} \) |
| 7 | \( 1 + 0.264T + 7T^{2} \) |
| 11 | \( 1 - 3.42T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 23 | \( 1 - 0.593T + 23T^{2} \) |
| 29 | \( 1 - 6.21T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 - 1.92T + 37T^{2} \) |
| 41 | \( 1 - 7.69T + 41T^{2} \) |
| 43 | \( 1 - 8.90T + 43T^{2} \) |
| 47 | \( 1 + 5.44T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 1.06T + 59T^{2} \) |
| 61 | \( 1 - 2.12T + 61T^{2} \) |
| 67 | \( 1 - 5.04T + 67T^{2} \) |
| 71 | \( 1 + 0.380T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 + 8.55T + 83T^{2} \) |
| 89 | \( 1 - 6.78T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.136993752147036418393659654835, −7.65764757124430436382480677217, −6.53784376627926595770763874699, −6.01968987498730988102596112128, −5.07456733710340284824919006701, −4.36324428245138008610924381863, −3.71051975529772234687174943824, −2.87649372002581599278863883153, −1.88287490484850075091985919425, −0.58747362515247987210303313695,
0.58747362515247987210303313695, 1.88287490484850075091985919425, 2.87649372002581599278863883153, 3.71051975529772234687174943824, 4.36324428245138008610924381863, 5.07456733710340284824919006701, 6.01968987498730988102596112128, 6.53784376627926595770763874699, 7.65764757124430436382480677217, 8.136993752147036418393659654835
