| L(s) = 1 | − 340.·2-s − 5.90e4·3-s − 1.98e6·4-s + 2.01e7·6-s + 6.73e8·7-s + 1.39e9·8-s + 3.48e9·9-s + 8.81e10·11-s + 1.16e11·12-s − 4.35e11·13-s − 2.29e11·14-s + 3.68e12·16-s + 3.38e12·17-s − 1.18e12·18-s − 4.68e13·19-s − 3.97e13·21-s − 3.00e13·22-s + 2.77e13·23-s − 8.20e13·24-s + 1.48e14·26-s − 2.05e14·27-s − 1.33e15·28-s − 4.93e14·29-s − 1.99e15·31-s − 4.17e15·32-s − 5.20e15·33-s − 1.15e15·34-s + ⋯ |
| L(s) = 1 | − 0.235·2-s − 0.577·3-s − 0.944·4-s + 0.135·6-s + 0.900·7-s + 0.457·8-s + 0.333·9-s + 1.02·11-s + 0.545·12-s − 0.877·13-s − 0.212·14-s + 0.836·16-s + 0.407·17-s − 0.0784·18-s − 1.75·19-s − 0.520·21-s − 0.241·22-s + 0.139·23-s − 0.264·24-s + 0.206·26-s − 0.192·27-s − 0.850·28-s − 0.217·29-s − 0.436·31-s − 0.654·32-s − 0.591·33-s − 0.0959·34-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 | \( 1 + 340.T + 2.09e6T^{2} \) |
| 7 | \( 1 - 6.73e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 8.81e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 4.35e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 3.38e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 4.68e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.77e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 4.93e14T + 5.13e30T^{2} \) |
| 31 | \( 1 + 1.99e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 2.48e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.53e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 8.38e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 1.06e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 2.83e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 7.02e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 6.54e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.27e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 9.18e18T + 7.52e38T^{2} \) |
| 73 | \( 1 + 3.67e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 6.42e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.62e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 2.98e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 6.86e20T + 5.27e41T^{2} \) |
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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
−10.08487898697867764316913755628, −9.041481195330474100747611465704, −8.110806324469742240421148935961, −6.92640247436663866891807056783, −5.59690471947318103903558907808, −4.63834832277891184665773100896, −3.88900399668348773016807928200, −2.03477515415093804182944642053, −1.00621754014060383817051786073, 0,
1.00621754014060383817051786073, 2.03477515415093804182944642053, 3.88900399668348773016807928200, 4.63834832277891184665773100896, 5.59690471947318103903558907808, 6.92640247436663866891807056783, 8.110806324469742240421148935961, 9.041481195330474100747611465704, 10.08487898697867764316913755628
