| L(s) = 1 | − 419.·2-s + 6.56e3·3-s + 4.48e4·4-s − 2.75e6·6-s − 1.07e7·7-s + 3.61e7·8-s + 4.30e7·9-s − 3.34e8·11-s + 2.94e8·12-s − 1.71e8·13-s + 4.49e9·14-s − 2.10e10·16-s + 3.73e9·17-s − 1.80e10·18-s + 3.46e10·19-s − 7.03e10·21-s + 1.40e11·22-s + 5.95e10·23-s + 2.37e11·24-s + 7.17e10·26-s + 2.82e11·27-s − 4.81e11·28-s − 2.08e12·29-s + 5.85e12·31-s + 4.08e12·32-s − 2.19e12·33-s − 1.56e12·34-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.577·3-s + 0.342·4-s − 0.668·6-s − 0.703·7-s + 0.762·8-s + 0.333·9-s − 0.470·11-s + 0.197·12-s − 0.0581·13-s + 0.814·14-s − 1.22·16-s + 0.129·17-s − 0.386·18-s + 0.468·19-s − 0.406·21-s + 0.544·22-s + 0.158·23-s + 0.440·24-s + 0.0673·26-s + 0.192·27-s − 0.240·28-s − 0.772·29-s + 1.23·31-s + 0.657·32-s − 0.271·33-s − 0.150·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 419.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 1.07e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 3.34e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.71e8T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.73e9T + 8.27e20T^{2} \) |
| 19 | \( 1 - 3.46e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 5.95e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.08e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 5.85e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.29e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 3.39e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 7.65e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 2.72e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 4.91e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 2.15e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.43e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 3.26e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 1.54e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.33e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 2.64e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.48e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 6.75e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 7.30e16T + 5.95e33T^{2} \) |
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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
−10.13551562274293342568762365046, −9.621189765328032863593024712222, −8.529102419773861499051003216970, −7.70638914782876624601584662331, −6.60702374640887259979126823789, −4.95390484751218972237609754696, −3.53718223166382404941165977836, −2.32368499369000765339989223412, −1.07431471315008776229912256911, 0,
1.07431471315008776229912256911, 2.32368499369000765339989223412, 3.53718223166382404941165977836, 4.95390484751218972237609754696, 6.60702374640887259979126823789, 7.70638914782876624601584662331, 8.529102419773861499051003216970, 9.621189765328032863593024712222, 10.13551562274293342568762365046
