| L(s) = 1 | − 15.8·2-s + 174.·3-s − 262.·4-s + 699.·5-s − 2.75e3·6-s − 1.40e3·7-s + 1.22e4·8-s + 1.06e4·9-s − 1.10e4·10-s + 7.56e3·11-s − 4.56e4·12-s − 3.42e4·13-s + 2.21e4·14-s + 1.21e5·15-s − 5.91e4·16-s + 1.64e5·17-s − 1.68e5·18-s − 7.52e5·19-s − 1.83e5·20-s − 2.44e5·21-s − 1.19e5·22-s + 1.17e6·23-s + 2.13e6·24-s − 1.46e6·25-s + 5.41e5·26-s − 1.57e6·27-s + 3.68e5·28-s + ⋯ |
| L(s) = 1 | − 0.698·2-s + 1.24·3-s − 0.512·4-s + 0.500·5-s − 0.867·6-s − 0.221·7-s + 1.05·8-s + 0.540·9-s − 0.349·10-s + 0.155·11-s − 0.635·12-s − 0.332·13-s + 0.154·14-s + 0.621·15-s − 0.225·16-s + 0.478·17-s − 0.377·18-s − 1.32·19-s − 0.256·20-s − 0.274·21-s − 0.108·22-s + 0.878·23-s + 1.31·24-s − 0.749·25-s + 0.232·26-s − 0.570·27-s + 0.113·28-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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + 1.50e9T \) |
| good | 2 | \( 1 + 15.8T + 512T^{2} \) |
| 3 | \( 1 - 174.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 699.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.40e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.56e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.42e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.64e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.52e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.17e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.17e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.13e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.65e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 6.73e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.62e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.33e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.72e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.63e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.15e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.16e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.39e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.71e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.60e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.23e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.99e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.49e8T + 7.60e17T^{2} \) |
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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
−9.841339309138855452192118555702, −9.368282489749752996243535333345, −8.424461730868837304495175202383, −7.80019962453857326600000273760, −6.44624868549961739229909438589, −4.91799142640203072149448999890, −3.72313108132378772953011715936, −2.51196907529283380861395825176, −1.42544825207447634080621965133, 0,
1.42544825207447634080621965133, 2.51196907529283380861395825176, 3.72313108132378772953011715936, 4.91799142640203072149448999890, 6.44624868549961739229909438589, 7.80019962453857326600000273760, 8.424461730868837304495175202383, 9.368282489749752996243535333345, 9.841339309138855452192118555702
