| L(s) = 1 | − 1.08e3·2-s + 5.90e4·3-s − 9.24e5·4-s − 6.39e7·6-s + 7.89e8·7-s + 3.27e9·8-s + 3.48e9·9-s + 1.26e11·11-s − 5.45e10·12-s + 8.10e10·13-s − 8.55e11·14-s − 1.60e12·16-s + 1.14e13·17-s − 3.77e12·18-s − 3.33e13·19-s + 4.66e13·21-s − 1.37e14·22-s − 2.69e14·23-s + 1.93e14·24-s − 8.77e13·26-s + 2.05e14·27-s − 7.29e14·28-s − 4.24e15·29-s − 4.38e15·31-s − 5.12e15·32-s + 7.47e15·33-s − 1.23e16·34-s + ⋯ |
| L(s) = 1 | − 0.747·2-s + 0.577·3-s − 0.440·4-s − 0.431·6-s + 1.05·7-s + 1.07·8-s + 0.333·9-s + 1.47·11-s − 0.254·12-s + 0.162·13-s − 0.790·14-s − 0.365·16-s + 1.37·17-s − 0.249·18-s − 1.24·19-s + 0.610·21-s − 1.10·22-s − 1.35·23-s + 0.622·24-s − 0.121·26-s + 0.192·27-s − 0.465·28-s − 1.87·29-s − 0.961·31-s − 0.804·32-s + 0.850·33-s − 1.02·34-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 | \( 1 + 1.08e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 7.89e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.26e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 8.10e10T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.14e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.33e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.69e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 4.24e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 4.38e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.00e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.19e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.96e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.56e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.03e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 4.27e17T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.02e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.24e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 3.25e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.51e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.33e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.80e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.47e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 5.24e20T + 5.27e41T^{2} \) |
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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
−9.825640494466150741809347797311, −8.977270987424500741775442460406, −8.163421021699561919502619057528, −7.35906814057784238234119786584, −5.79128850178709659308864338109, −4.38357169622133145656963111772, −3.69964449523213420946613145616, −1.82203510002846300864999146810, −1.36616930218782869467384846106, 0,
1.36616930218782869467384846106, 1.82203510002846300864999146810, 3.69964449523213420946613145616, 4.38357169622133145656963111772, 5.79128850178709659308864338109, 7.35906814057784238234119786584, 8.163421021699561919502619057528, 8.977270987424500741775442460406, 9.825640494466150741809347797311
