| L(s) = 1 | + 2.27·2-s + 3.18·4-s − 2.98·5-s − 2.44·7-s + 2.70·8-s − 6.79·10-s + 3.22·11-s − 2.73·13-s − 5.57·14-s − 0.220·16-s + 2.14·17-s + 1.46·19-s − 9.50·20-s + 7.34·22-s − 23-s + 3.90·25-s − 6.22·26-s − 7.79·28-s + 29-s + 6.98·31-s − 5.90·32-s + 4.87·34-s + 7.30·35-s + 10.5·37-s + 3.34·38-s − 8.06·40-s + 6.90·41-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.59·4-s − 1.33·5-s − 0.924·7-s + 0.955·8-s − 2.14·10-s + 0.972·11-s − 0.757·13-s − 1.48·14-s − 0.0551·16-s + 0.519·17-s + 0.336·19-s − 2.12·20-s + 1.56·22-s − 0.208·23-s + 0.781·25-s − 1.21·26-s − 1.47·28-s + 0.185·29-s + 1.25·31-s − 1.04·32-s + 0.836·34-s + 1.23·35-s + 1.73·37-s + 0.542·38-s − 1.27·40-s + 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.349039287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.349039287\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 2.14T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 + 9.50T + 43T^{2} \) |
| 47 | \( 1 - 3.28T + 47T^{2} \) |
| 53 | \( 1 - 7.20T + 53T^{2} \) |
| 59 | \( 1 + 7.50T + 59T^{2} \) |
| 61 | \( 1 + 0.996T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 5.24T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 + 7.48T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 + 7.50T + 89T^{2} \) |
| 97 | \( 1 - 8.72T + 97T^{2} \) |
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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
−7.74820438062170697978140271380, −7.20786949609322791460467937090, −6.43523720402318168833763163656, −6.00582733870149493661996363906, −4.92396414214451123199737829866, −4.39291877086772120673551994812, −3.68426259686321645886976272816, −3.21934084416034007828951289850, −2.36902789891962741390020402709, −0.73466472838324786519669804511,
0.73466472838324786519669804511, 2.36902789891962741390020402709, 3.21934084416034007828951289850, 3.68426259686321645886976272816, 4.39291877086772120673551994812, 4.92396414214451123199737829866, 6.00582733870149493661996363906, 6.43523720402318168833763163656, 7.20786949609322791460467937090, 7.74820438062170697978140271380
