| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 0.414·11-s − 12-s − 13-s − 15-s + 16-s + 2.41·17-s − 18-s − 1.82·19-s + 20-s + 0.414·22-s + 2.65·23-s + 24-s − 4·25-s + 26-s − 27-s − 8.65·29-s + 30-s + 4.24·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.124·11-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 0.250·16-s + 0.585·17-s − 0.235·18-s − 0.419·19-s + 0.223·20-s + 0.0883·22-s + 0.553·23-s + 0.204·24-s − 0.800·25-s + 0.196·26-s − 0.192·27-s − 1.60·29-s + 0.182·30-s + 0.762·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 + 1.82T + 19T^{2} \) |
| 23 | \( 1 - 2.65T + 23T^{2} \) |
| 29 | \( 1 + 8.65T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 + 8.41T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 + 9.07T + 71T^{2} \) |
| 73 | \( 1 + 6.17T + 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 - 0.343T + 83T^{2} \) |
| 89 | \( 1 - 3.41T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.046255284652506278349440589318, −7.47717444341696420356416605910, −6.66553271887956183356488485778, −5.95235842153120318319577140052, −5.33735176862639062986958347826, −4.38211573493227544096717014405, −3.30474825443667136373169267406, −2.23975415598967972535426682463, −1.31397148507771552339153423959, 0,
1.31397148507771552339153423959, 2.23975415598967972535426682463, 3.30474825443667136373169267406, 4.38211573493227544096717014405, 5.33735176862639062986958347826, 5.95235842153120318319577140052, 6.66553271887956183356488485778, 7.47717444341696420356416605910, 8.046255284652506278349440589318
