L(s) = 1 | + 2.86·3-s − 0.0527·5-s − 1.89·7-s + 5.19·9-s − 5.46·11-s − 0.847·13-s − 0.151·15-s − 17-s + 0.788·19-s − 5.42·21-s + 0.154·23-s − 4.99·25-s + 6.29·27-s − 3.17·29-s − 8.43·31-s − 15.6·33-s + 0.100·35-s − 3.87·37-s − 2.42·39-s − 1.73·41-s − 2.07·43-s − 0.274·45-s + 3.26·47-s − 3.40·49-s − 2.86·51-s + 2.52·53-s + 0.288·55-s + ⋯ |
L(s) = 1 | + 1.65·3-s − 0.0236·5-s − 0.716·7-s + 1.73·9-s − 1.64·11-s − 0.235·13-s − 0.0390·15-s − 0.242·17-s + 0.181·19-s − 1.18·21-s + 0.0322·23-s − 0.999·25-s + 1.21·27-s − 0.588·29-s − 1.51·31-s − 2.72·33-s + 0.0169·35-s − 0.637·37-s − 0.388·39-s − 0.270·41-s − 0.316·43-s − 0.0409·45-s + 0.475·47-s − 0.487·49-s − 0.400·51-s + 0.346·53-s + 0.0389·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 2.86T + 3T^{2} \) |
| 5 | \( 1 + 0.0527T + 5T^{2} \) |
| 7 | \( 1 + 1.89T + 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 + 0.847T + 13T^{2} \) |
| 19 | \( 1 - 0.788T + 19T^{2} \) |
| 23 | \( 1 - 0.154T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 8.43T + 31T^{2} \) |
| 37 | \( 1 + 3.87T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 + 2.07T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 2.52T + 53T^{2} \) |
| 61 | \( 1 - 8.67T + 61T^{2} \) |
| 67 | \( 1 + 6.26T + 67T^{2} \) |
| 71 | \( 1 - 1.81T + 71T^{2} \) |
| 73 | \( 1 - 4.69T + 73T^{2} \) |
| 79 | \( 1 - 3.52T + 79T^{2} \) |
| 83 | \( 1 + 6.16T + 83T^{2} \) |
| 89 | \( 1 + 8.42T + 89T^{2} \) |
| 97 | \( 1 - 6.52T + 97T^{2} \) |
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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
−8.082067733053969298351763296607, −7.51410075862935183523890680076, −6.97171510187242170993076114729, −5.79481497243760817669362594423, −5.03490317891756573592136895756, −3.91120787083792045548731595635, −3.32511985107230159937620994411, −2.53745503599766375077134988684, −1.88214908499253841231632912509, 0,
1.88214908499253841231632912509, 2.53745503599766375077134988684, 3.32511985107230159937620994411, 3.91120787083792045548731595635, 5.03490317891756573592136895756, 5.79481497243760817669362594423, 6.97171510187242170993076114729, 7.51410075862935183523890680076, 8.082067733053969298351763296607