L(s) = 1 | + 9·3-s + 14.0·5-s + 106.·7-s + 81·9-s − 94.6·11-s − 158.·13-s + 126.·15-s − 9.93·17-s − 1.58e3·19-s + 957.·21-s + 529·23-s − 2.92e3·25-s + 729·27-s + 6.55e3·29-s − 2.79e3·31-s − 851.·33-s + 1.49e3·35-s − 9.30e3·37-s − 1.42e3·39-s − 715.·41-s − 1.86e4·43-s + 1.13e3·45-s + 8.11e3·47-s − 5.48e3·49-s − 89.3·51-s − 7.25e3·53-s − 1.33e3·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.251·5-s + 0.820·7-s + 0.333·9-s − 0.235·11-s − 0.260·13-s + 0.145·15-s − 0.00833·17-s − 1.00·19-s + 0.473·21-s + 0.208·23-s − 0.936·25-s + 0.192·27-s + 1.44·29-s − 0.521·31-s − 0.136·33-s + 0.206·35-s − 1.11·37-s − 0.150·39-s − 0.0664·41-s − 1.53·43-s + 0.0838·45-s + 0.535·47-s − 0.326·49-s − 0.00481·51-s − 0.354·53-s − 0.0593·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 23 | \( 1 - 529T \) |
good | 5 | \( 1 - 14.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 106.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 94.6T + 1.61e5T^{2} \) |
| 13 | \( 1 + 158.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 9.93T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.58e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 6.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.30e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 715.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.86e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.11e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.25e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.85e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.38e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.91e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.31e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.62e4T + 8.58e9T^{2} \) |
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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
−8.554338847987884840537539541283, −8.081396288659113284479315746235, −7.14267032157833651085510280517, −6.24361058410726256590747829679, −5.13127696794858796146558406531, −4.41492967611958527784955188637, −3.30213582021830651369258405782, −2.22658718772388420829084791838, −1.46682723180141736791860070569, 0,
1.46682723180141736791860070569, 2.22658718772388420829084791838, 3.30213582021830651369258405782, 4.41492967611958527784955188637, 5.13127696794858796146558406531, 6.24361058410726256590747829679, 7.14267032157833651085510280517, 8.081396288659113284479315746235, 8.554338847987884840537539541283