| L(s) = 1 | + 525.·2-s − 5.29e3·3-s + 1.45e5·4-s + 1.28e6·5-s − 2.78e6·6-s − 1.26e7·7-s + 7.56e6·8-s − 1.01e8·9-s + 6.77e8·10-s − 3.28e8·11-s − 7.69e8·12-s − 6.61e8·13-s − 6.67e9·14-s − 6.81e9·15-s − 1.50e10·16-s − 2.15e10·17-s − 5.31e10·18-s + 1.53e10·19-s + 1.87e11·20-s + 6.71e10·21-s − 1.72e11·22-s − 8.81e10·23-s − 4.00e10·24-s + 8.98e11·25-s − 3.48e11·26-s + 1.21e12·27-s − 1.84e12·28-s + ⋯ |
| L(s) = 1 | + 1.45·2-s − 0.465·3-s + 1.10·4-s + 1.47·5-s − 0.676·6-s − 0.831·7-s + 0.159·8-s − 0.783·9-s + 2.14·10-s − 0.462·11-s − 0.516·12-s − 0.225·13-s − 1.20·14-s − 0.686·15-s − 0.878·16-s − 0.747·17-s − 1.13·18-s + 0.207·19-s + 1.63·20-s + 0.387·21-s − 0.671·22-s − 0.234·23-s − 0.0742·24-s + 1.17·25-s − 0.326·26-s + 0.830·27-s − 0.922·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 5.00e11T \) |
| good | 2 | \( 1 - 525.T + 1.31e5T^{2} \) |
| 3 | \( 1 + 5.29e3T + 1.29e8T^{2} \) |
| 5 | \( 1 - 1.28e6T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.26e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 3.28e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 6.61e8T + 8.65e18T^{2} \) |
| 17 | \( 1 + 2.15e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.53e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 8.81e10T + 1.41e23T^{2} \) |
| 31 | \( 1 + 7.26e11T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.73e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 4.59e12T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.20e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 5.32e12T + 2.66e28T^{2} \) |
| 53 | \( 1 - 3.52e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.46e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 4.73e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 5.23e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.52e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 5.74e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 8.13e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.94e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 2.34e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.50e17T + 5.95e33T^{2} \) |
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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
−13.04570876711809884607875438555, −11.82022564771985706606797021028, −10.39229506867664979796334846962, −9.081551089340272542483612336068, −6.61747525397900006001364464690, −5.83869309627869537788859893222, −4.97454740539236693504864231362, −3.19796926457314289337711397250, −2.14740236059084016063782333820, 0,
2.14740236059084016063782333820, 3.19796926457314289337711397250, 4.97454740539236693504864231362, 5.83869309627869537788859893222, 6.61747525397900006001364464690, 9.081551089340272542483612336068, 10.39229506867664979796334846962, 11.82022564771985706606797021028, 13.04570876711809884607875438555
