| L(s) = 1 | + 2.48·3-s + 1.00·5-s + 0.658·7-s + 3.16·9-s − 5.12·11-s − 2.19·13-s + 2.50·15-s − 4.93·17-s + 19-s + 1.63·21-s + 2.25·23-s − 3.98·25-s + 0.403·27-s − 3.27·29-s − 6.68·31-s − 12.7·33-s + 0.664·35-s + 4.75·37-s − 5.43·39-s − 6.29·41-s − 7.36·43-s + 3.19·45-s − 8.45·47-s − 6.56·49-s − 12.2·51-s + 6.21·53-s − 5.17·55-s + ⋯ |
| L(s) = 1 | + 1.43·3-s + 0.451·5-s + 0.248·7-s + 1.05·9-s − 1.54·11-s − 0.607·13-s + 0.646·15-s − 1.19·17-s + 0.229·19-s + 0.356·21-s + 0.469·23-s − 0.796·25-s + 0.0776·27-s − 0.608·29-s − 1.20·31-s − 2.21·33-s + 0.112·35-s + 0.781·37-s − 0.870·39-s − 0.983·41-s − 1.12·43-s + 0.475·45-s − 1.23·47-s − 0.938·49-s − 1.71·51-s + 0.853·53-s − 0.697·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 - 1.00T + 5T^{2} \) |
| 7 | \( 1 - 0.658T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 23 | \( 1 - 2.25T + 23T^{2} \) |
| 29 | \( 1 + 3.27T + 29T^{2} \) |
| 31 | \( 1 + 6.68T + 31T^{2} \) |
| 37 | \( 1 - 4.75T + 37T^{2} \) |
| 41 | \( 1 + 6.29T + 41T^{2} \) |
| 43 | \( 1 + 7.36T + 43T^{2} \) |
| 47 | \( 1 + 8.45T + 47T^{2} \) |
| 53 | \( 1 - 6.21T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 - 1.82T + 61T^{2} \) |
| 67 | \( 1 - 7.94T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 3.62T + 73T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 8.58T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78614465283573301758761031123, −7.31378119464755716667948325583, −6.44408503117723342050137312764, −5.35551473171815897735114299153, −4.90941014308574698461909747886, −3.84759552254540565072523535097, −3.07812707033569525885183708608, −2.28073760498055780276754522943, −1.86181145829630040651669316719, 0,
1.86181145829630040651669316719, 2.28073760498055780276754522943, 3.07812707033569525885183708608, 3.84759552254540565072523535097, 4.90941014308574698461909747886, 5.35551473171815897735114299153, 6.44408503117723342050137312764, 7.31378119464755716667948325583, 7.78614465283573301758761031123
