| L(s) = 1 | − 4.76·2-s − 3·3-s + 14.6·4-s − 5·5-s + 14.2·6-s + 35.2·7-s − 31.8·8-s + 9·9-s + 23.8·10-s − 41.1·11-s − 44.0·12-s − 79.0·13-s − 167.·14-s + 15·15-s + 34.1·16-s + 27.4·17-s − 42.8·18-s + 142.·19-s − 73.4·20-s − 105.·21-s + 196.·22-s − 125.·23-s + 95.4·24-s + 25·25-s + 376.·26-s − 27·27-s + 517.·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.577·3-s + 1.83·4-s − 0.447·5-s + 0.972·6-s + 1.90·7-s − 1.40·8-s + 0.333·9-s + 0.753·10-s − 1.12·11-s − 1.05·12-s − 1.68·13-s − 3.20·14-s + 0.258·15-s + 0.533·16-s + 0.391·17-s − 0.561·18-s + 1.72·19-s − 0.820·20-s − 1.09·21-s + 1.89·22-s − 1.13·23-s + 0.812·24-s + 0.200·25-s + 2.83·26-s − 0.192·27-s + 3.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 4.76T + 8T^{2} \) |
| 7 | \( 1 - 35.2T + 343T^{2} \) |
| 11 | \( 1 + 41.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 27.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 125.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 63.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 39.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 385.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 534.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 505.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 184.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 56.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 313.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 163.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 695.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 571.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 573.T + 9.12e5T^{2} \) |
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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
−10.22194392280061065018941467545, −9.513348498867647610739672898011, −8.094729364441957749078802615091, −7.83421917774038893596698102036, −7.15928592817148718621916201143, −5.43137721193709529404816428607, −4.69255868965361302978212719829, −2.49340377944610569412060072768, −1.29947420673431791649719015475, 0,
1.29947420673431791649719015475, 2.49340377944610569412060072768, 4.69255868965361302978212719829, 5.43137721193709529404816428607, 7.15928592817148718621916201143, 7.83421917774038893596698102036, 8.094729364441957749078802615091, 9.513348498867647610739672898011, 10.22194392280061065018941467545
