| L(s) = 1 | + 3.11·2-s + 5.69·4-s + 2.14·5-s + 1.36i·7-s + 5.26·8-s + 6.69·10-s + 0.221·11-s + (−7.69 − 10.4i)13-s + 4.24i·14-s − 6.38·16-s + 24.1i·17-s − 27.7i·19-s + 12.2·20-s + 0.690·22-s + 28.3i·23-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 1.42·4-s + 0.429·5-s + 0.194i·7-s + 0.657·8-s + 0.669·10-s + 0.0201·11-s + (−0.591 − 0.806i)13-s + 0.303i·14-s − 0.398·16-s + 1.42i·17-s − 1.46i·19-s + 0.611·20-s + 0.0313·22-s + 1.23i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0175i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.98805 + 0.0261843i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.98805 + 0.0261843i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (7.69 + 10.4i)T \) |
| good | 2 | \( 1 - 3.11T + 4T^{2} \) |
| 5 | \( 1 - 2.14T + 25T^{2} \) |
| 7 | \( 1 - 1.36iT - 49T^{2} \) |
| 11 | \( 1 - 0.221T + 121T^{2} \) |
| 17 | \( 1 - 24.1iT - 289T^{2} \) |
| 19 | \( 1 + 27.7iT - 361T^{2} \) |
| 23 | \( 1 - 28.3iT - 529T^{2} \) |
| 29 | \( 1 + 27.0iT - 841T^{2} \) |
| 31 | \( 1 - 18.6iT - 961T^{2} \) |
| 37 | \( 1 + 20.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 14.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 30.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 78.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 26.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 52.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 11.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 76.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 75.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 122. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 62.6T + 6.24e3T^{2} \) |
| 83 | \( 1 - 94.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 21.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 133. iT - 9.40e3T^{2} \) |
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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
−13.31877583908627315468010619950, −12.56523624105233828567446465600, −11.58666457941826692669616063875, −10.44145265362339689744019777533, −9.081841383559654982407255426569, −7.46598879426445458531261358213, −6.09753907390752052988824869088, −5.27871167316522094180107909537, −3.91255435858274359424700093379, −2.44814460435717312872983340000,
2.40645258341439104543050965701, 3.98391998312522841863972510343, 5.12256823743093219097407264140, 6.24847405864604929947245678453, 7.36547685809360030384950955445, 9.135039432155504654463147018735, 10.36241275898755473043373895081, 11.73363893406068270662478091773, 12.32661153554343913745469682390, 13.50106410878804419539122704473
