| L(s) = 1 | − 145.·2-s − 2.02e3·3-s + 1.28e4·4-s − 1.01e4·5-s + 2.93e5·6-s + 3.45e5·7-s − 6.80e5·8-s + 2.50e6·9-s + 1.47e6·10-s + 1.29e6·11-s − 2.60e7·12-s − 2.93e7·13-s − 5.02e7·14-s + 2.05e7·15-s − 6.76e6·16-s − 7.66e7·17-s − 3.63e8·18-s + 1.13e8·19-s − 1.30e8·20-s − 7.00e8·21-s − 1.87e8·22-s − 9.76e8·23-s + 1.37e9·24-s − 1.11e9·25-s + 4.25e9·26-s − 1.84e9·27-s + 4.45e9·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 1.60·3-s + 1.57·4-s − 0.290·5-s + 2.57·6-s + 1.11·7-s − 0.917·8-s + 1.57·9-s + 0.466·10-s + 0.220·11-s − 2.52·12-s − 1.68·13-s − 1.78·14-s + 0.466·15-s − 0.100·16-s − 0.770·17-s − 2.52·18-s + 0.555·19-s − 0.457·20-s − 1.78·21-s − 0.353·22-s − 1.37·23-s + 1.47·24-s − 0.915·25-s + 2.70·26-s − 0.917·27-s + 1.74·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.07668838194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07668838194\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - 5.84e13T \) |
| good | 2 | \( 1 + 145.T + 8.19e3T^{2} \) |
| 3 | \( 1 + 2.02e3T + 1.59e6T^{2} \) |
| 5 | \( 1 + 1.01e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 3.45e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.29e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.93e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 7.66e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 1.13e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 9.76e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 4.14e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 5.83e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 3.87e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.31e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.34e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.16e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 3.23e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 4.58e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 7.04e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 7.66e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.10e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.36e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 7.79e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 6.47e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 1.00e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.52e13T + 6.73e25T^{2} \) |
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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
−10.26185144741292670980411419279, −9.387721139966086537224948162031, −8.083485074107388433836465787506, −7.39351013308023232175737017808, −6.47336620009903149360724138120, −5.21557589687898099310241384254, −4.40417537251106014500315838701, −2.17992291487395160363956825499, −1.32949828037724494854108753722, −0.17334210609360390726615418409,
0.17334210609360390726615418409, 1.32949828037724494854108753722, 2.17992291487395160363956825499, 4.40417537251106014500315838701, 5.21557589687898099310241384254, 6.47336620009903149360724138120, 7.39351013308023232175737017808, 8.083485074107388433836465787506, 9.387721139966086537224948162031, 10.26185144741292670980411419279
