| L(s) = 1 | + 12.5·2-s − 253.·3-s − 354.·4-s − 210.·5-s − 3.18e3·6-s + 2.90e3·7-s − 1.08e4·8-s + 4.45e4·9-s − 2.64e3·10-s + 1.02e4·11-s + 8.98e4·12-s − 5.79e4·13-s + 3.64e4·14-s + 5.34e4·15-s + 4.48e4·16-s − 3.21e5·17-s + 5.59e5·18-s − 3.35e5·19-s + 7.46e4·20-s − 7.36e5·21-s + 1.28e5·22-s − 7.39e5·23-s + 2.75e6·24-s − 1.90e6·25-s − 7.27e5·26-s − 6.30e6·27-s − 1.02e6·28-s + ⋯ |
| L(s) = 1 | + 0.554·2-s − 1.80·3-s − 0.692·4-s − 0.150·5-s − 1.00·6-s + 0.457·7-s − 0.938·8-s + 2.26·9-s − 0.0836·10-s + 0.211·11-s + 1.25·12-s − 0.562·13-s + 0.253·14-s + 0.272·15-s + 0.171·16-s − 0.932·17-s + 1.25·18-s − 0.591·19-s + 0.104·20-s − 0.826·21-s + 0.117·22-s − 0.550·23-s + 1.69·24-s − 0.977·25-s − 0.312·26-s − 2.28·27-s − 0.316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2076896795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2076896795\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - 1.50e9T \) |
| good | 2 | \( 1 - 12.5T + 512T^{2} \) |
| 3 | \( 1 + 253.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 210.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.90e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.02e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.79e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.21e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.35e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.39e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.29e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.89e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.15e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.35e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.90e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.44e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.73e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.95e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.33e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.64e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.21e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.35e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.74e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.81e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.94e8T + 7.60e17T^{2} \) |
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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
−11.09748827148410241856536896506, −10.10496719621982818352976006392, −9.005623493001606137353683352679, −7.52566836742219174602773743562, −6.34591308759721477698575912407, −5.51267254118182925797575528350, −4.68443850678266884506517520321, −3.90838162622562184096199671835, −1.75977059373394720982647911509, −0.22046956627293151567152698738,
0.22046956627293151567152698738, 1.75977059373394720982647911509, 3.90838162622562184096199671835, 4.68443850678266884506517520321, 5.51267254118182925797575528350, 6.34591308759721477698575912407, 7.52566836742219174602773743562, 9.005623493001606137353683352679, 10.10496719621982818352976006392, 11.09748827148410241856536896506
