| L(s) = 1 | − 2.84·2-s + 24.3·3-s − 23.8·4-s + 56.3·5-s − 69.3·6-s − 117.·7-s + 159.·8-s + 349.·9-s − 160.·10-s − 541.·11-s − 581.·12-s − 714.·13-s + 334.·14-s + 1.37e3·15-s + 310.·16-s + 809.·17-s − 994.·18-s − 2.00e3·19-s − 1.34e3·20-s − 2.85e3·21-s + 1.54e3·22-s − 2.29e3·23-s + 3.87e3·24-s + 49.0·25-s + 2.03e3·26-s + 2.58e3·27-s + 2.80e3·28-s + ⋯ |
| L(s) = 1 | − 0.503·2-s + 1.56·3-s − 0.746·4-s + 1.00·5-s − 0.786·6-s − 0.904·7-s + 0.879·8-s + 1.43·9-s − 0.507·10-s − 1.34·11-s − 1.16·12-s − 1.17·13-s + 0.455·14-s + 1.57·15-s + 0.303·16-s + 0.678·17-s − 0.723·18-s − 1.27·19-s − 0.752·20-s − 1.41·21-s + 0.679·22-s − 0.905·23-s + 1.37·24-s + 0.0156·25-s + 0.590·26-s + 0.681·27-s + 0.675·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + 3.88e4T \) |
| good | 2 | \( 1 + 2.84T + 32T^{2} \) |
| 3 | \( 1 - 24.3T + 243T^{2} \) |
| 5 | \( 1 - 56.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 117.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 541.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 714.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 809.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 941.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 403.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.80e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.18e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.49e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.07e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.69e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.43e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.23e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.05e4T + 8.58e9T^{2} \) |
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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
−10.38680974529768165865357404104, −9.810390494357405342854801316126, −9.293081050931040945831355593997, −8.192470223961550194803512842385, −7.50412045715680576265232246934, −5.83837054647567262851163058295, −4.40466855159649229725166089988, −2.94430906180933895981245094783, −2.01444151775574973203847094348, 0,
2.01444151775574973203847094348, 2.94430906180933895981245094783, 4.40466855159649229725166089988, 5.83837054647567262851163058295, 7.50412045715680576265232246934, 8.192470223961550194803512842385, 9.293081050931040945831355593997, 9.810390494357405342854801316126, 10.38680974529768165865357404104
