| L(s) = 1 | − 7.91·2-s − 12.8·3-s + 30.6·4-s + 101.·6-s + 172.·7-s + 11.0·8-s − 77.1·9-s + 452.·11-s − 394.·12-s + 22.7·13-s − 1.36e3·14-s − 1.06e3·16-s + 521.·17-s + 610.·18-s + 1.55e3·19-s − 2.21e3·21-s − 3.57e3·22-s + 3.46e3·23-s − 142.·24-s − 179.·26-s + 4.12e3·27-s + 5.27e3·28-s + 4.32e3·29-s − 3.98e3·31-s + 8.08e3·32-s − 5.82e3·33-s − 4.12e3·34-s + ⋯ |
| L(s) = 1 | − 1.39·2-s − 0.826·3-s + 0.956·4-s + 1.15·6-s + 1.32·7-s + 0.0610·8-s − 0.317·9-s + 1.12·11-s − 0.790·12-s + 0.0373·13-s − 1.85·14-s − 1.04·16-s + 0.437·17-s + 0.443·18-s + 0.990·19-s − 1.09·21-s − 1.57·22-s + 1.36·23-s − 0.0504·24-s − 0.0522·26-s + 1.08·27-s + 1.27·28-s + 0.954·29-s − 0.745·31-s + 1.39·32-s − 0.931·33-s − 0.612·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 + 7.91T + 32T^{2} \) |
| 3 | \( 1 + 12.8T + 243T^{2} \) |
| 7 | \( 1 - 172.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 452.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 22.7T + 3.71e5T^{2} \) |
| 17 | \( 1 - 521.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.46e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.00e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.64e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 2.41e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.30e4T + 8.58e9T^{2} \) |
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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
−8.739613279340522586731719540037, −8.127863504902508818637814155826, −7.20265635108620104260190793056, −6.49890518481028173012930518757, −5.25298071066134720789170384931, −4.71444070145950000531443069486, −3.20327314298884996069483560442, −1.57527765769530933093251075514, −1.16413680985575281826891191551, 0,
1.16413680985575281826891191551, 1.57527765769530933093251075514, 3.20327314298884996069483560442, 4.71444070145950000531443069486, 5.25298071066134720789170384931, 6.49890518481028173012930518757, 7.20265635108620104260190793056, 8.127863504902508818637814155826, 8.739613279340522586731719540037
