| L(s) = 1 | − 5·2-s − 4·3-s + 17·4-s + 5·5-s + 20·6-s − 32·7-s − 45·8-s − 11·9-s − 25·10-s − 12·11-s − 68·12-s + 42·13-s + 160·14-s − 20·15-s + 89·16-s + 114·17-s + 55·18-s + 85·20-s + 128·21-s + 60·22-s + 160·23-s + 180·24-s + 25·25-s − 210·26-s + 152·27-s − 544·28-s − 214·29-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 0.769·3-s + 17/8·4-s + 0.447·5-s + 1.36·6-s − 1.72·7-s − 1.98·8-s − 0.407·9-s − 0.790·10-s − 0.328·11-s − 1.63·12-s + 0.896·13-s + 3.05·14-s − 0.344·15-s + 1.39·16-s + 1.62·17-s + 0.720·18-s + 0.950·20-s + 1.33·21-s + 0.581·22-s + 1.45·23-s + 1.53·24-s + 1/5·25-s − 1.58·26-s + 1.08·27-s − 3.67·28-s − 1.37·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4211109654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4211109654\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 23 | \( 1 - 160 T + p^{3} T^{2} \) |
| 29 | \( 1 + 214 T + p^{3} T^{2} \) |
| 31 | \( 1 - 144 T + p^{3} T^{2} \) |
| 37 | \( 1 + 94 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 308 T + p^{3} T^{2} \) |
| 47 | \( 1 - 184 T + p^{3} T^{2} \) |
| 53 | \( 1 - 274 T + p^{3} T^{2} \) |
| 59 | \( 1 + 276 T + p^{3} T^{2} \) |
| 61 | \( 1 + 826 T + p^{3} T^{2} \) |
| 67 | \( 1 + 52 T + p^{3} T^{2} \) |
| 71 | \( 1 - 344 T + p^{3} T^{2} \) |
| 73 | \( 1 + 166 T + p^{3} T^{2} \) |
| 79 | \( 1 - 688 T + p^{3} T^{2} \) |
| 83 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 89 | \( 1 + 1578 T + p^{3} T^{2} \) |
| 97 | \( 1 + 786 T + p^{3} T^{2} \) |
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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
−9.152090677196926133792827999632, −8.306621272836001695634926346011, −7.36191885521070931948794726126, −6.60682794417736458502232480846, −6.04028987463467789232148086825, −5.34158654215574349926925012512, −3.40879592304262666344884897971, −2.76828599452867951232272590928, −1.29797326150678463510011530167, −0.45902853590718972909651517644,
0.45902853590718972909651517644, 1.29797326150678463510011530167, 2.76828599452867951232272590928, 3.40879592304262666344884897971, 5.34158654215574349926925012512, 6.04028987463467789232148086825, 6.60682794417736458502232480846, 7.36191885521070931948794726126, 8.306621272836001695634926346011, 9.152090677196926133792827999632
