| L(s) = 1 | + 1.71·2-s − 3-s + 0.936·4-s − 4.13·5-s − 1.71·6-s + 0.967·7-s − 1.82·8-s + 9-s − 7.08·10-s − 6.54·11-s − 0.936·12-s + 2.28·13-s + 1.65·14-s + 4.13·15-s − 4.99·16-s − 2.37·17-s + 1.71·18-s + 2.94·19-s − 3.87·20-s − 0.967·21-s − 11.2·22-s − 1.52·23-s + 1.82·24-s + 12.0·25-s + 3.91·26-s − 27-s + 0.906·28-s + ⋯ |
| L(s) = 1 | + 1.21·2-s − 0.577·3-s + 0.468·4-s − 1.84·5-s − 0.699·6-s + 0.365·7-s − 0.644·8-s + 0.333·9-s − 2.24·10-s − 1.97·11-s − 0.270·12-s + 0.632·13-s + 0.443·14-s + 1.06·15-s − 1.24·16-s − 0.576·17-s + 0.403·18-s + 0.674·19-s − 0.866·20-s − 0.211·21-s − 2.39·22-s − 0.317·23-s + 0.371·24-s + 2.41·25-s + 0.767·26-s − 0.192·27-s + 0.171·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.71T + 2T^{2} \) |
| 5 | \( 1 + 4.13T + 5T^{2} \) |
| 7 | \( 1 - 0.967T + 7T^{2} \) |
| 11 | \( 1 + 6.54T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 - 2.59T + 29T^{2} \) |
| 31 | \( 1 + 4.67T + 31T^{2} \) |
| 37 | \( 1 + 9.81T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 3.66T + 59T^{2} \) |
| 61 | \( 1 + 0.791T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 6.43T + 71T^{2} \) |
| 73 | \( 1 + 3.78T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 + 7.18T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−11.16180428303573038224507577517, −10.55377371901416929093408841397, −8.789666520429239281463038806206, −7.87961542583986394501519462937, −7.07187014920186412227047950691, −5.64916929760519503679072174404, −4.85681173389904397463110875212, −4.02655611392867995851540816621, −2.96600652691672338147029552614, 0,
2.96600652691672338147029552614, 4.02655611392867995851540816621, 4.85681173389904397463110875212, 5.64916929760519503679072174404, 7.07187014920186412227047950691, 7.87961542583986394501519462937, 8.789666520429239281463038806206, 10.55377371901416929093408841397, 11.16180428303573038224507577517
