| L(s) = 1 | − 2.91·2-s + 8.67·3-s + 0.491·4-s − 25.2·6-s − 6.75·7-s + 21.8·8-s + 48.1·9-s + 13.0·11-s + 4.26·12-s + 5.65·13-s + 19.6·14-s − 67.6·16-s + 31.9·17-s − 140.·18-s + 4.86·19-s − 58.5·21-s − 38.1·22-s + 6.08·23-s + 189.·24-s − 16.4·26-s + 183.·27-s − 3.32·28-s + 152.·29-s + 229.·31-s + 22.2·32-s + 113.·33-s − 93.1·34-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 1.66·3-s + 0.0614·4-s − 1.71·6-s − 0.364·7-s + 0.966·8-s + 1.78·9-s + 0.358·11-s + 0.102·12-s + 0.120·13-s + 0.375·14-s − 1.05·16-s + 0.456·17-s − 1.83·18-s + 0.0587·19-s − 0.608·21-s − 0.369·22-s + 0.0551·23-s + 1.61·24-s − 0.124·26-s + 1.30·27-s − 0.0224·28-s + 0.977·29-s + 1.33·31-s + 0.122·32-s + 0.598·33-s − 0.470·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.226910260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.226910260\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 + 2.91T + 8T^{2} \) |
| 3 | \( 1 - 8.67T + 27T^{2} \) |
| 7 | \( 1 + 6.75T + 343T^{2} \) |
| 11 | \( 1 - 13.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.65T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.86T + 6.85e3T^{2} \) |
| 23 | \( 1 - 6.08T + 1.21e4T^{2} \) |
| 29 | \( 1 - 152.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 265.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 72.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 67.8T + 7.95e4T^{2} \) |
| 53 | \( 1 - 217.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 493.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 479.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 783.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 777.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 189.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 59.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 445.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 112.T + 9.12e5T^{2} \) |
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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
−9.208599000456095307231577401257, −8.646626137866986532045145673209, −8.076189683000186485956745253754, −7.34138234574273529718798341858, −6.47203939495970398434306370950, −4.87976700381345321725453173426, −3.89485172644743397624540147314, −3.01579143549557931505186769697, −1.92999329761861620749639336475, −0.865232307418060411460440901111,
0.865232307418060411460440901111, 1.92999329761861620749639336475, 3.01579143549557931505186769697, 3.89485172644743397624540147314, 4.87976700381345321725453173426, 6.47203939495970398434306370950, 7.34138234574273529718798341858, 8.076189683000186485956745253754, 8.646626137866986532045145673209, 9.208599000456095307231577401257
