| L(s) = 1 | + 0.223·2-s − 81·3-s − 511.·4-s − 18.1·6-s + 580.·7-s − 228.·8-s + 6.56e3·9-s + 4.72e4·11-s + 4.14e4·12-s − 2.23e4·13-s + 129.·14-s + 2.62e5·16-s + 2.44e5·17-s + 1.46e3·18-s − 2.48e5·19-s − 4.70e4·21-s + 1.05e4·22-s + 9.90e5·23-s + 1.85e4·24-s − 5.00e3·26-s − 5.31e5·27-s − 2.97e5·28-s − 4.35e6·29-s − 5.60e6·31-s + 1.75e5·32-s − 3.82e6·33-s + 5.47e4·34-s + ⋯ |
| L(s) = 1 | + 0.00988·2-s − 0.577·3-s − 0.999·4-s − 0.00570·6-s + 0.0914·7-s − 0.0197·8-s + 0.333·9-s + 0.973·11-s + 0.577·12-s − 0.217·13-s + 0.000903·14-s + 0.999·16-s + 0.711·17-s + 0.00329·18-s − 0.437·19-s − 0.0527·21-s + 0.00962·22-s + 0.737·23-s + 0.0114·24-s − 0.00214·26-s − 0.192·27-s − 0.0914·28-s − 1.14·29-s − 1.08·31-s + 0.0296·32-s − 0.562·33-s + 0.00702·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.223T + 512T^{2} \) |
| 7 | \( 1 - 580.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.72e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.23e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.44e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.48e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.90e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.35e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.60e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.73e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 6.11e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.89e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.09e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.16e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.79e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.24e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.90e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.13e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.03e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.94e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.52e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.25e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.33e8T + 7.60e17T^{2} \) |
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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
−12.21679192464115826217409094422, −11.03452212236800007408737314554, −9.767613821976772256823903962579, −8.865397676097852454433052591669, −7.43915316306565752672792062755, −5.96226130616545840510892575049, −4.80203846455339891811580044803, −3.61111577737741892798413844383, −1.36364476965959026948754584161, 0,
1.36364476965959026948754584161, 3.61111577737741892798413844383, 4.80203846455339891811580044803, 5.96226130616545840510892575049, 7.43915316306565752672792062755, 8.865397676097852454433052591669, 9.767613821976772256823903962579, 11.03452212236800007408737314554, 12.21679192464115826217409094422
