| L(s) = 1 | + 810.·2-s − 2.29e4·3-s + 1.32e5·4-s + 1.95e6·5-s − 1.85e7·6-s − 1.14e8·7-s − 3.17e8·8-s − 6.35e8·9-s + 1.58e9·10-s + 1.95e9·11-s − 3.03e9·12-s − 4.04e10·13-s − 9.29e10·14-s − 4.48e10·15-s − 3.26e11·16-s − 1.49e11·17-s − 5.15e11·18-s + 1.82e12·19-s + 2.58e11·20-s + 2.63e12·21-s + 1.58e12·22-s − 6.34e11·23-s + 7.28e12·24-s + 3.81e12·25-s − 3.27e13·26-s + 4.12e13·27-s − 1.51e13·28-s + ⋯ |
| L(s) = 1 | + 1.11·2-s − 0.673·3-s + 0.252·4-s + 0.447·5-s − 0.753·6-s − 1.07·7-s − 0.836·8-s − 0.547·9-s + 0.500·10-s + 0.249·11-s − 0.169·12-s − 1.05·13-s − 1.20·14-s − 0.300·15-s − 1.18·16-s − 0.305·17-s − 0.612·18-s + 1.29·19-s + 0.112·20-s + 0.723·21-s + 0.279·22-s − 0.0734·23-s + 0.562·24-s + 0.199·25-s − 1.18·26-s + 1.04·27-s − 0.271·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 1.95e6T \) |
| good | 2 | \( 1 - 810.T + 5.24e5T^{2} \) |
| 3 | \( 1 + 2.29e4T + 1.16e9T^{2} \) |
| 7 | \( 1 + 1.14e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.95e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 4.04e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 1.49e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.82e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 6.34e11T + 7.46e25T^{2} \) |
| 29 | \( 1 - 6.07e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 2.16e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.45e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 2.25e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 2.75e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 1.33e16T + 5.88e31T^{2} \) |
| 53 | \( 1 + 2.39e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 2.17e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.30e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 3.37e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 5.03e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 3.64e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.30e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.00e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 2.64e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 4.87e18T + 5.60e37T^{2} \) |
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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
−17.95999843394244911624330642767, −16.41745501307115528598446610168, −14.54841325185355813549005298009, −13.10791958330124846812503920470, −11.80035455163403749369677636900, −9.557910684594814793278624939829, −6.41375764722231202742940310756, −5.11395785348767624181801561293, −3.05484137371130244124996458687, 0,
3.05484137371130244124996458687, 5.11395785348767624181801561293, 6.41375764722231202742940310756, 9.557910684594814793278624939829, 11.80035455163403749369677636900, 13.10791958330124846812503920470, 14.54841325185355813549005298009, 16.41745501307115528598446610168, 17.95999843394244911624330642767
