| L(s) = 1 | + 2.70e3·2-s − 1.29e5·3-s − 1.09e6·4-s + 4.88e7·5-s − 3.49e8·6-s + 5.55e9·7-s − 2.56e10·8-s − 7.74e10·9-s + 1.31e11·10-s − 1.73e12·11-s + 1.41e11·12-s − 2.03e12·13-s + 1.50e13·14-s − 6.31e12·15-s − 6.00e13·16-s + 8.51e13·17-s − 2.09e14·18-s − 9.35e14·19-s − 5.32e13·20-s − 7.18e14·21-s − 4.69e15·22-s − 1.42e15·23-s + 3.31e15·24-s + 2.38e15·25-s − 5.49e15·26-s + 2.21e16·27-s − 6.05e15·28-s + ⋯ |
| L(s) = 1 | + 0.932·2-s − 0.421·3-s − 0.130·4-s + 0.447·5-s − 0.393·6-s + 1.06·7-s − 1.05·8-s − 0.822·9-s + 0.417·10-s − 1.83·11-s + 0.0548·12-s − 0.314·13-s + 0.990·14-s − 0.188·15-s − 0.853·16-s + 0.602·17-s − 0.766·18-s − 1.84·19-s − 0.0581·20-s − 0.447·21-s − 1.71·22-s − 0.311·23-s + 0.444·24-s + 0.200·25-s − 0.293·26-s + 0.768·27-s − 0.138·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 4.88e7T \) |
| good | 2 | \( 1 - 2.70e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 1.29e5T + 9.41e10T^{2} \) |
| 7 | \( 1 - 5.55e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.73e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 2.03e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 8.51e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 9.35e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 1.42e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 4.05e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 8.62e15T + 2.00e34T^{2} \) |
| 37 | \( 1 - 2.94e17T + 1.17e36T^{2} \) |
| 41 | \( 1 + 5.73e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 4.86e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 3.10e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 1.06e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.80e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 3.12e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.10e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 1.31e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 2.55e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 9.38e20T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.44e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 6.53e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 7.92e22T + 4.96e45T^{2} \) |
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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
−17.23691760897384912906453781860, −15.07641405689013413350955401138, −13.83337364403607488605640018873, −12.39037570666784245071830470186, −10.67308614501053613510628283000, −8.338295243056947317669193501559, −5.76440815429138473858841946767, −4.76694092180680855717465167691, −2.53985022148423492135862621662, 0,
2.53985022148423492135862621662, 4.76694092180680855717465167691, 5.76440815429138473858841946767, 8.338295243056947317669193501559, 10.67308614501053613510628283000, 12.39037570666784245071830470186, 13.83337364403607488605640018873, 15.07641405689013413350955401138, 17.23691760897384912906453781860
