| L(s) = 1 | + 2.22·2-s − 3.07·4-s − 14.7·5-s + 26.0·7-s − 24.5·8-s − 32.6·10-s + 0.340·11-s − 88.6·13-s + 57.8·14-s − 30.0·16-s + 78.2·17-s + 83.2·19-s + 45.1·20-s + 0.755·22-s + 74.9·23-s + 91.5·25-s − 196.·26-s − 80.0·28-s + 244.·29-s + 101.·31-s + 130.·32-s + 173.·34-s − 383.·35-s − 408.·37-s + 184.·38-s + 361.·40-s − 378.·41-s + ⋯ |
| L(s) = 1 | + 0.784·2-s − 0.383·4-s − 1.31·5-s + 1.40·7-s − 1.08·8-s − 1.03·10-s + 0.00933·11-s − 1.89·13-s + 1.10·14-s − 0.468·16-s + 1.11·17-s + 1.00·19-s + 0.505·20-s + 0.00732·22-s + 0.679·23-s + 0.732·25-s − 1.48·26-s − 0.539·28-s + 1.56·29-s + 0.585·31-s + 0.718·32-s + 0.876·34-s − 1.85·35-s − 1.81·37-s + 0.788·38-s + 1.42·40-s − 1.44·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 2.22T + 8T^{2} \) |
| 5 | \( 1 + 14.7T + 125T^{2} \) |
| 7 | \( 1 - 26.0T + 343T^{2} \) |
| 11 | \( 1 - 0.340T + 1.33e3T^{2} \) |
| 13 | \( 1 + 88.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 78.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 83.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 74.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 244.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 408.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 378.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 124.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 469.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 198.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 683.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 786.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 292.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 551.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 623.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 630.T + 9.12e5T^{2} \) |
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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
−8.186175498137430731286083959830, −7.60497920510268827511236855247, −6.96987722459208957169099092957, −5.49991158334174658346449138170, −4.90056107156855950176980583755, −4.52756341065982364318715785753, −3.48570912130871323694638664685, −2.71437279943513936278428743054, −1.14459017841055415296648796025, 0,
1.14459017841055415296648796025, 2.71437279943513936278428743054, 3.48570912130871323694638664685, 4.52756341065982364318715785753, 4.90056107156855950176980583755, 5.49991158334174658346449138170, 6.96987722459208957169099092957, 7.60497920510268827511236855247, 8.186175498137430731286083959830
