| L(s) = 1 | + 31.9·2-s + 506.·4-s − 1.63e3·5-s + 9.37e3·7-s − 177.·8-s − 5.21e4·10-s − 74.3·11-s − 2.85e4·13-s + 2.99e5·14-s − 2.64e5·16-s − 2.10e4·17-s − 9.67e5·19-s − 8.28e5·20-s − 2.37e3·22-s + 1.49e6·23-s + 7.22e5·25-s − 9.11e5·26-s + 4.74e6·28-s + 7.44e5·29-s − 5.33e6·31-s − 8.36e6·32-s − 6.72e5·34-s − 1.53e7·35-s − 1.21e7·37-s − 3.08e7·38-s + 2.90e5·40-s + 1.06e7·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.989·4-s − 1.17·5-s + 1.47·7-s − 0.0153·8-s − 1.65·10-s − 0.00153·11-s − 0.277·13-s + 2.08·14-s − 1.01·16-s − 0.0612·17-s − 1.70·19-s − 1.15·20-s − 0.00216·22-s + 1.11·23-s + 0.369·25-s − 0.391·26-s + 1.46·28-s + 0.195·29-s − 1.03·31-s − 1.41·32-s − 0.0863·34-s − 1.72·35-s − 1.06·37-s − 2.40·38-s + 0.0179·40-s + 0.588·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 - 31.9T + 512T^{2} \) |
| 5 | \( 1 + 1.63e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.37e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 74.3T + 2.35e9T^{2} \) |
| 17 | \( 1 + 2.10e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.67e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.49e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.44e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.33e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.21e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.06e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.31e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.10e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.32e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.70e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.05e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.17e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.05e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 8.93e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 8.48e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.41e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.29e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.66e8T + 7.60e17T^{2} \) |
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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
−11.47075465292176968363282110410, −10.84561727408536642982128716651, −8.796272172215031836474411432759, −7.84690148808459190211136384266, −6.64491396162887946814348396506, −5.09441152839897224961790429079, −4.47706398241931041920243857030, −3.42524621144022122251053826039, −1.90660955126249441691964062889, 0,
1.90660955126249441691964062889, 3.42524621144022122251053826039, 4.47706398241931041920243857030, 5.09441152839897224961790429079, 6.64491396162887946814348396506, 7.84690148808459190211136384266, 8.796272172215031836474411432759, 10.84561727408536642982128716651, 11.47075465292176968363282110410
