| L(s) = 1 | + 1.41·2-s − 3-s − 5-s − 1.41·6-s − 7-s − 2.82·8-s + 9-s − 1.41·10-s + 11-s + 3.41·13-s − 1.41·14-s + 15-s − 4.00·16-s + 3.82·17-s + 1.41·18-s + 1.82·19-s + 21-s + 1.41·22-s + 1.82·23-s + 2.82·24-s + 25-s + 4.82·26-s − 27-s + 7.24·29-s + 1.41·30-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.577·3-s − 0.447·5-s − 0.577·6-s − 0.377·7-s − 0.999·8-s + 0.333·9-s − 0.447·10-s + 0.301·11-s + 0.946·13-s − 0.377·14-s + 0.258·15-s − 1.00·16-s + 0.928·17-s + 0.333·18-s + 0.419·19-s + 0.218·21-s + 0.301·22-s + 0.381·23-s + 0.577·24-s + 0.200·25-s + 0.946·26-s − 0.192·27-s + 1.34·29-s + 0.258·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.786580530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.786580530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 3.82T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 - 7.24T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 3.07T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 2.65T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 0.656T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 0.242T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 1.24T + 97T^{2} \) |
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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
−9.900582590763942512380926127677, −8.966388619124155708189401254919, −8.160467642174212165676544948325, −6.98963571312339861262062748431, −6.21706954108423051533508463304, −5.49734949891421853928581309596, −4.59286799291106978557856143906, −3.75935684225144076739711708886, −2.95138526931300296985986346485, −0.926626054464907288495888226959,
0.926626054464907288495888226959, 2.95138526931300296985986346485, 3.75935684225144076739711708886, 4.59286799291106978557856143906, 5.49734949891421853928581309596, 6.21706954108423051533508463304, 6.98963571312339861262062748431, 8.160467642174212165676544948325, 8.966388619124155708189401254919, 9.900582590763942512380926127677
