| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 10.7·5-s − 6·6-s − 3.65·7-s + 8·8-s + 9·9-s − 21.4·10-s + 27.0·11-s − 12·12-s + 20.1·13-s − 7.30·14-s + 32.1·15-s + 16·16-s − 85.6·17-s + 18·18-s + 120.·19-s − 42.8·20-s + 10.9·21-s + 54.1·22-s + 50.4·23-s − 24·24-s − 10.2·25-s + 40.3·26-s − 27·27-s − 14.6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.958·5-s − 0.408·6-s − 0.197·7-s + 0.353·8-s + 0.333·9-s − 0.677·10-s + 0.742·11-s − 0.288·12-s + 0.430·13-s − 0.139·14-s + 0.553·15-s + 0.250·16-s − 1.22·17-s + 0.235·18-s + 1.44·19-s − 0.479·20-s + 0.113·21-s + 0.525·22-s + 0.457·23-s − 0.204·24-s − 0.0816·25-s + 0.304·26-s − 0.192·27-s − 0.0986·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 113 | \( 1 + 113T \) |
| good | 5 | \( 1 + 10.7T + 125T^{2} \) |
| 7 | \( 1 + 3.65T + 343T^{2} \) |
| 11 | \( 1 - 27.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 85.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 50.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 292.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 23.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 390.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 146.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 1.06T + 7.95e4T^{2} \) |
| 47 | \( 1 + 161.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 208.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 647.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 289.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 313.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 59.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 84.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 183.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 387.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 744.T + 9.12e5T^{2} \) |
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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
−9.722263872897702992043103483497, −8.808508211056949990021317265614, −7.59353161391527972287321733284, −6.94059997719564787152066566220, −6.00838594548838131986845514140, −5.00671342156883146847347316447, −4.03540958742229895058325261105, −3.28786863754030857093667626400, −1.56260599289597163395183360885, 0,
1.56260599289597163395183360885, 3.28786863754030857093667626400, 4.03540958742229895058325261105, 5.00671342156883146847347316447, 6.00838594548838131986845514140, 6.94059997719564787152066566220, 7.59353161391527972287321733284, 8.808508211056949990021317265614, 9.722263872897702992043103483497
