| L(s) = 1 | − 1.92·2-s + 1.71·4-s − 2.16·5-s − 2.75·7-s + 0.554·8-s + 4.17·10-s − 11-s − 2.10·13-s + 5.30·14-s − 4.49·16-s + 2.21·17-s + 0.816·19-s − 3.70·20-s + 1.92·22-s − 3.38·23-s − 0.308·25-s + 4.04·26-s − 4.71·28-s + 0.749·29-s − 2.25·31-s + 7.54·32-s − 4.27·34-s + 5.96·35-s + 5.07·37-s − 1.57·38-s − 1.20·40-s − 8.11·41-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 0.856·4-s − 0.968·5-s − 1.04·7-s + 0.196·8-s + 1.31·10-s − 0.301·11-s − 0.582·13-s + 1.41·14-s − 1.12·16-s + 0.538·17-s + 0.187·19-s − 0.829·20-s + 0.410·22-s − 0.705·23-s − 0.0616·25-s + 0.794·26-s − 0.890·28-s + 0.139·29-s − 0.405·31-s + 1.33·32-s − 0.733·34-s + 1.00·35-s + 0.833·37-s − 0.255·38-s − 0.189·40-s − 1.26·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6633 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1492462574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1492462574\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 + 1.92T + 2T^{2} \) |
| 5 | \( 1 + 2.16T + 5T^{2} \) |
| 7 | \( 1 + 2.75T + 7T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 17 | \( 1 - 2.21T + 17T^{2} \) |
| 19 | \( 1 - 0.816T + 19T^{2} \) |
| 23 | \( 1 + 3.38T + 23T^{2} \) |
| 29 | \( 1 - 0.749T + 29T^{2} \) |
| 31 | \( 1 + 2.25T + 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 + 7.31T + 43T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 - 7.95T + 59T^{2} \) |
| 61 | \( 1 - 0.926T + 61T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 8.53T + 73T^{2} \) |
| 79 | \( 1 - 1.31T + 79T^{2} \) |
| 83 | \( 1 + 6.99T + 83T^{2} \) |
| 89 | \( 1 + 6.33T + 89T^{2} \) |
| 97 | \( 1 + 5.49T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.184649071379034317490813773063, −7.37690098157288762302295209435, −7.00633845928254651595432986301, −6.13937757624042408829258546625, −5.17216807551942937638696094552, −4.26255722806137591574179003684, −3.48060830835727680850042557290, −2.64487711721075872024119828856, −1.51179816391379529958392256412, −0.24805613712840239665050095237,
0.24805613712840239665050095237, 1.51179816391379529958392256412, 2.64487711721075872024119828856, 3.48060830835727680850042557290, 4.26255722806137591574179003684, 5.17216807551942937638696094552, 6.13937757624042408829258546625, 7.00633845928254651595432986301, 7.37690098157288762302295209435, 8.184649071379034317490813773063
