| L(s) = 1 | − 3.60·2-s + 3·3-s + 4.97·4-s + 5·5-s − 10.8·6-s − 31.6·7-s + 10.9·8-s + 9·9-s − 18.0·10-s + 14.9·12-s − 38.8·13-s + 113.·14-s + 15·15-s − 79.0·16-s − 138.·17-s − 32.4·18-s − 12.5·19-s + 24.8·20-s − 94.8·21-s − 179.·23-s + 32.7·24-s + 25·25-s + 139.·26-s + 27·27-s − 157.·28-s − 119.·29-s − 54.0·30-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.577·3-s + 0.621·4-s + 0.447·5-s − 0.735·6-s − 1.70·7-s + 0.482·8-s + 0.333·9-s − 0.569·10-s + 0.358·12-s − 0.828·13-s + 2.17·14-s + 0.258·15-s − 1.23·16-s − 1.97·17-s − 0.424·18-s − 0.152·19-s + 0.277·20-s − 0.985·21-s − 1.62·23-s + 0.278·24-s + 0.200·25-s + 1.05·26-s + 0.192·27-s − 1.06·28-s − 0.762·29-s − 0.328·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2815110143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2815110143\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 3.60T + 8T^{2} \) |
| 7 | \( 1 + 31.6T + 343T^{2} \) |
| 13 | \( 1 + 38.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 138.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 179.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 47.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 74.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 375.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 76.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 866.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 442.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 633.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 484.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 436.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 472.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 249.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 236.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.52e3T + 9.12e5T^{2} \) |
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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
−9.038695503761967112167953727993, −8.398475857088818391303033327545, −7.41288154938285360421059337825, −6.74964830240632535477376975754, −6.10893661138658899342970194791, −4.69054480888648824539252254453, −3.75033995049429060257948107802, −2.54611547018043129420590280641, −1.90064964020114025610781533867, −0.27604139538996374723181536907,
0.27604139538996374723181536907, 1.90064964020114025610781533867, 2.54611547018043129420590280641, 3.75033995049429060257948107802, 4.69054480888648824539252254453, 6.10893661138658899342970194791, 6.74964830240632535477376975754, 7.41288154938285360421059337825, 8.398475857088818391303033327545, 9.038695503761967112167953727993
