| L(s) = 1 | + 582.·2-s − 6.56e3·3-s + 2.08e5·4-s − 3.82e6·6-s + 8.47e6·7-s + 4.52e7·8-s + 4.30e7·9-s − 8.02e8·11-s − 1.36e9·12-s + 6.02e8·13-s + 4.94e9·14-s − 9.56e8·16-s + 3.92e9·17-s + 2.50e10·18-s + 8.27e10·19-s − 5.56e10·21-s − 4.67e11·22-s − 4.91e11·23-s − 2.97e11·24-s + 3.51e11·26-s − 2.82e11·27-s + 1.76e12·28-s + 1.29e12·29-s − 3.99e12·31-s − 6.49e12·32-s + 5.26e12·33-s + 2.28e12·34-s + ⋯ |
| L(s) = 1 | + 1.61·2-s − 0.577·3-s + 1.59·4-s − 0.929·6-s + 0.555·7-s + 0.954·8-s + 0.333·9-s − 1.12·11-s − 0.919·12-s + 0.204·13-s + 0.894·14-s − 0.0556·16-s + 0.136·17-s + 0.536·18-s + 1.11·19-s − 0.320·21-s − 1.81·22-s − 1.30·23-s − 0.551·24-s + 0.329·26-s − 0.192·27-s + 0.885·28-s + 0.480·29-s − 0.840·31-s − 1.04·32-s + 0.651·33-s + 0.219·34-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 | \( 1 - 582.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 8.47e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 8.02e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 6.02e8T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.92e9T + 8.27e20T^{2} \) |
| 19 | \( 1 - 8.27e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 4.91e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 1.29e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 3.99e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 8.00e11T + 4.56e26T^{2} \) |
| 41 | \( 1 + 6.03e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 2.29e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.45e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 1.16e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 2.51e12T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.11e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.68e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 4.61e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.16e16T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.21e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.18e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 5.38e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 9.94e16T + 5.95e33T^{2} \) |
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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
−11.15133023847088856302204715591, −9.984102706995082048734331826910, −8.111651821524728354852822992352, −6.93151827106451310032188210555, −5.67071915140388877261496696916, −5.12643995033363892760029125343, −4.02988302144466538161840821554, −2.86103409762132679496860779058, −1.63093325135112445079894090366, 0,
1.63093325135112445079894090366, 2.86103409762132679496860779058, 4.02988302144466538161840821554, 5.12643995033363892760029125343, 5.67071915140388877261496696916, 6.93151827106451310032188210555, 8.111651821524728354852822992352, 9.984102706995082048734331826910, 11.15133023847088856302204715591
