| L(s) = 1 | + 7.35·2-s + 42.6·3-s − 457.·4-s − 1.23e3·5-s + 313.·6-s + 892.·7-s − 7.13e3·8-s − 1.78e4·9-s − 9.09e3·10-s − 2.71e4·11-s − 1.95e4·12-s − 2.85e4·13-s + 6.56e3·14-s − 5.26e4·15-s + 1.81e5·16-s − 3.46e4·17-s − 1.31e5·18-s + 4.28e5·19-s + 5.66e5·20-s + 3.80e4·21-s − 1.99e5·22-s + 2.03e6·23-s − 3.04e5·24-s − 4.24e5·25-s − 2.10e5·26-s − 1.60e6·27-s − 4.08e5·28-s + ⋯ |
| L(s) = 1 | + 0.325·2-s + 0.303·3-s − 0.894·4-s − 0.884·5-s + 0.0987·6-s + 0.140·7-s − 0.615·8-s − 0.907·9-s − 0.287·10-s − 0.559·11-s − 0.271·12-s − 0.277·13-s + 0.0456·14-s − 0.268·15-s + 0.694·16-s − 0.100·17-s − 0.295·18-s + 0.755·19-s + 0.791·20-s + 0.0426·21-s − 0.181·22-s + 1.51·23-s − 0.187·24-s − 0.217·25-s − 0.0901·26-s − 0.579·27-s − 0.125·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 7.35T + 512T^{2} \) |
| 3 | \( 1 - 42.6T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.23e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 892.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.71e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.46e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.28e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.03e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.26e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.15e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 7.58e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.92e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.71e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.95e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.72e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.13e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.76e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.90e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.87e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.61e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.42e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.80e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.59e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.46e9T + 7.60e17T^{2} \) |
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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
−17.07393907198354444207628956339, −15.33256819579566625066153488249, −14.22505092003998841012283411943, −12.85445491453793972570809894225, −11.31969655516604555701088514367, −9.196526681358649258536784772497, −7.82905030775083749862309007042, −5.19631160721890757832532363900, −3.38532871414856603989847282361, 0,
3.38532871414856603989847282361, 5.19631160721890757832532363900, 7.82905030775083749862309007042, 9.196526681358649258536784772497, 11.31969655516604555701088514367, 12.85445491453793972570809894225, 14.22505092003998841012283411943, 15.33256819579566625066153488249, 17.07393907198354444207628956339
