| L(s) = 1 | + 2-s − 3-s + 4-s − 0.722·5-s − 6-s + 8-s + 9-s − 0.722·10-s + 1.13·11-s − 12-s − 1.27·13-s + 0.722·15-s + 16-s + 3.93·17-s + 18-s − 2·19-s − 0.722·20-s + 1.13·22-s − 23-s − 24-s − 4.47·25-s − 1.27·26-s − 27-s − 5.76·29-s + 0.722·30-s − 6.20·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.323·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.228·10-s + 0.341·11-s − 0.288·12-s − 0.354·13-s + 0.186·15-s + 0.250·16-s + 0.954·17-s + 0.235·18-s − 0.458·19-s − 0.161·20-s + 0.241·22-s − 0.208·23-s − 0.204·24-s − 0.895·25-s − 0.250·26-s − 0.192·27-s − 1.07·29-s + 0.131·30-s − 1.11·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 0.722T + 5T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 5.76T + 29T^{2} \) |
| 31 | \( 1 + 6.20T + 31T^{2} \) |
| 37 | \( 1 - 6.08T + 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 + 3.01T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 0.290T + 73T^{2} \) |
| 79 | \( 1 - 0.290T + 79T^{2} \) |
| 83 | \( 1 - 8.42T + 83T^{2} \) |
| 89 | \( 1 + 0.0631T + 89T^{2} \) |
| 97 | \( 1 + 0.750T + 97T^{2} \) |
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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
−7.66294424305391222648562516571, −6.77658072219514281485973760402, −6.10105958852863133789494858518, −5.49917491340100099267445805055, −4.81552737757104401961765908020, −3.96149356923393788835711542590, −3.47297623075049083184693589518, −2.31497319533247266143115321025, −1.39341535780073484769676425262, 0,
1.39341535780073484769676425262, 2.31497319533247266143115321025, 3.47297623075049083184693589518, 3.96149356923393788835711542590, 4.81552737757104401961765908020, 5.49917491340100099267445805055, 6.10105958852863133789494858518, 6.77658072219514281485973760402, 7.66294424305391222648562516571
