| L(s) = 1 | − 125.·2-s + 179.·3-s + 7.46e3·4-s − 2.67e4·5-s − 2.24e4·6-s + 1.07e5·7-s + 9.12e4·8-s − 1.56e6·9-s + 3.34e6·10-s + 8.88e6·11-s + 1.33e6·12-s + 2.92e7·13-s − 1.34e7·14-s − 4.79e6·15-s − 7.25e7·16-s − 2.41e7·17-s + 1.95e8·18-s − 3.44e8·19-s − 1.99e8·20-s + 1.91e7·21-s − 1.11e9·22-s + 5.61e7·23-s + 1.63e7·24-s − 5.05e8·25-s − 3.65e9·26-s − 5.65e8·27-s + 7.99e8·28-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.141·3-s + 0.910·4-s − 0.765·5-s − 0.196·6-s + 0.344·7-s + 0.123·8-s − 0.979·9-s + 1.05·10-s + 1.51·11-s + 0.129·12-s + 1.67·13-s − 0.475·14-s − 0.108·15-s − 1.08·16-s − 0.242·17-s + 1.35·18-s − 1.67·19-s − 0.697·20-s + 0.0488·21-s − 2.08·22-s + 0.0790·23-s + 0.0174·24-s − 0.413·25-s − 2.32·26-s − 0.280·27-s + 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + 2.41e7T \) |
| good | 2 | \( 1 + 125.T + 8.19e3T^{2} \) |
| 3 | \( 1 - 179.T + 1.59e6T^{2} \) |
| 5 | \( 1 + 2.67e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 1.07e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 8.88e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 2.92e7T + 3.02e14T^{2} \) |
| 19 | \( 1 + 3.44e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 5.61e7T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.90e8T + 1.02e19T^{2} \) |
| 31 | \( 1 - 1.06e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.47e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 5.87e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.19e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 9.33e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 6.79e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 2.41e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 2.13e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 9.95e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 4.92e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.10e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.53e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.36e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 4.84e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.27e13T + 6.73e25T^{2} \) |
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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
−15.36677106225776120257865222958, −13.86815089254324623939885851717, −11.70598851753120348863406902751, −10.78685009734487378259920088466, −8.861876637429140912298576745965, −8.340445101490666630758277847492, −6.54457124743331549326287534130, −3.91436207667576542548305281839, −1.54311094827835722858519178260, 0,
1.54311094827835722858519178260, 3.91436207667576542548305281839, 6.54457124743331549326287534130, 8.340445101490666630758277847492, 8.861876637429140912298576745965, 10.78685009734487378259920088466, 11.70598851753120348863406902751, 13.86815089254324623939885851717, 15.36677106225776120257865222958
