| L(s) = 1 | − 1.52·2-s + 3-s + 0.318·4-s − 1.52·6-s − 0.990·7-s + 2.56·8-s + 9-s + 5.97·11-s + 0.318·12-s + 4.02·13-s + 1.50·14-s − 4.53·16-s + 0.476·17-s − 1.52·18-s + 2.82·19-s − 0.990·21-s − 9.09·22-s − 1.74·23-s + 2.56·24-s − 6.13·26-s + 27-s − 0.315·28-s − 1.41·29-s + 8.76·31-s + 1.78·32-s + 5.97·33-s − 0.725·34-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.577·3-s + 0.159·4-s − 0.621·6-s − 0.374·7-s + 0.905·8-s + 0.333·9-s + 1.80·11-s + 0.0918·12-s + 1.11·13-s + 0.403·14-s − 1.13·16-s + 0.115·17-s − 0.358·18-s + 0.647·19-s − 0.216·21-s − 1.93·22-s − 0.364·23-s + 0.522·24-s − 1.20·26-s + 0.192·27-s − 0.0596·28-s − 0.263·29-s + 1.57·31-s + 0.315·32-s + 1.03·33-s − 0.124·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.322201766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322201766\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + 1.52T + 2T^{2} \) |
| 7 | \( 1 + 0.990T + 7T^{2} \) |
| 11 | \( 1 - 5.97T + 11T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 - 0.476T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 8.76T + 31T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 + 5.50T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 + 9.62T + 47T^{2} \) |
| 53 | \( 1 - 6.57T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 5.43T + 71T^{2} \) |
| 73 | \( 1 + 4.08T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 2.29T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−9.149380948135333689580358200722, −8.523333306902368452326369081114, −8.068994008288444731720190506914, −6.82965534504925837809423139769, −6.54800146501137124639923884537, −5.14476284157472806151080061262, −3.97833354630058450278201646203, −3.42775251076730872544131980689, −1.82740767757816540804184779295, −0.975061769768781866501334334119,
0.975061769768781866501334334119, 1.82740767757816540804184779295, 3.42775251076730872544131980689, 3.97833354630058450278201646203, 5.14476284157472806151080061262, 6.54800146501137124639923884537, 6.82965534504925837809423139769, 8.068994008288444731720190506914, 8.523333306902368452326369081114, 9.149380948135333689580358200722
