| L(s) = 1 | + 0.532·2-s + 1.85·3-s − 7.71·4-s + 18.1·5-s + 0.988·6-s + 20.2·7-s − 8.37·8-s − 23.5·9-s + 9.64·10-s + 11·11-s − 14.3·12-s + 10.8·14-s + 33.6·15-s + 57.2·16-s − 66.4·17-s − 12.5·18-s − 139.·19-s − 139.·20-s + 37.6·21-s + 5.86·22-s − 30.4·23-s − 15.5·24-s + 202.·25-s − 93.8·27-s − 156.·28-s + 195.·29-s + 17.9·30-s + ⋯ |
| L(s) = 1 | + 0.188·2-s + 0.357·3-s − 0.964·4-s + 1.61·5-s + 0.0672·6-s + 1.09·7-s − 0.370·8-s − 0.872·9-s + 0.305·10-s + 0.301·11-s − 0.344·12-s + 0.206·14-s + 0.578·15-s + 0.894·16-s − 0.948·17-s − 0.164·18-s − 1.68·19-s − 1.56·20-s + 0.391·21-s + 0.0567·22-s − 0.275·23-s − 0.132·24-s + 1.62·25-s − 0.668·27-s − 1.05·28-s + 1.25·29-s + 0.108·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.532T + 8T^{2} \) |
| 3 | \( 1 - 1.85T + 27T^{2} \) |
| 5 | \( 1 - 18.1T + 125T^{2} \) |
| 7 | \( 1 - 20.2T + 343T^{2} \) |
| 17 | \( 1 + 66.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 30.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 195.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 70.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 216.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 450.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 582.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 351.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 129.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 153.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 138.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 377.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
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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
−8.611620941415685685006847012149, −8.146217684891525502104414736281, −6.55111956913665250288896714156, −6.10512449117016303292693027027, −4.98429422832274366031819310232, −4.72154812770468310254670668500, −3.38828307542788076516618552438, −2.24694286120300030161491895360, −1.56313149749515230405419583940, 0,
1.56313149749515230405419583940, 2.24694286120300030161491895360, 3.38828307542788076516618552438, 4.72154812770468310254670668500, 4.98429422832274366031819310232, 6.10512449117016303292693027027, 6.55111956913665250288896714156, 8.146217684891525502104414736281, 8.611620941415685685006847012149
