| L(s) = 1 | + 594.·2-s − 5.90e4·3-s − 1.74e6·4-s − 3.50e7·6-s − 9.49e8·7-s − 2.28e9·8-s + 3.48e9·9-s − 8.95e10·11-s + 1.02e11·12-s − 1.19e11·13-s − 5.64e11·14-s + 2.30e12·16-s − 7.66e11·17-s + 2.07e12·18-s + 1.00e13·19-s + 5.60e13·21-s − 5.32e13·22-s + 1.37e14·23-s + 1.34e14·24-s − 7.07e13·26-s − 2.05e14·27-s + 1.65e15·28-s + 5.32e14·29-s − 2.67e15·31-s + 6.15e15·32-s + 5.28e15·33-s − 4.55e14·34-s + ⋯ |
| L(s) = 1 | + 0.410·2-s − 0.577·3-s − 0.831·4-s − 0.236·6-s − 1.27·7-s − 0.751·8-s + 0.333·9-s − 1.04·11-s + 0.480·12-s − 0.239·13-s − 0.521·14-s + 0.523·16-s − 0.0922·17-s + 0.136·18-s + 0.376·19-s + 0.733·21-s − 0.427·22-s + 0.692·23-s + 0.433·24-s − 0.0982·26-s − 0.192·27-s + 1.05·28-s + 0.234·29-s − 0.585·31-s + 0.966·32-s + 0.601·33-s − 0.0378·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 594.T + 2.09e6T^{2} \) |
| 7 | \( 1 + 9.49e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 8.95e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 1.19e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 7.66e11T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.00e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.37e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 5.32e14T + 5.13e30T^{2} \) |
| 31 | \( 1 + 2.67e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.68e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.01e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 6.08e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 2.05e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.16e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.09e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 4.71e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.22e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.77e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.43e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 8.41e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 5.17e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.29e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.15e21T + 5.27e41T^{2} \) |
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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
−9.973174887393218717471283734719, −9.318624386645748152444531133632, −7.921598740890459372466383964911, −6.61914546543645434390501295080, −5.62512437972718003963078218491, −4.78059883177774675264258757892, −3.59260968897993316390598011360, −2.64280519250350795021789583125, −0.794658356326118763510337156067, 0,
0.794658356326118763510337156067, 2.64280519250350795021789583125, 3.59260968897993316390598011360, 4.78059883177774675264258757892, 5.62512437972718003963078218491, 6.61914546543645434390501295080, 7.921598740890459372466383964911, 9.318624386645748152444531133632, 9.973174887393218717471283734719
