| L(s) = 1 | + 2.45·2-s − 3-s + 4.04·4-s − 5-s − 2.45·6-s − 0.945·7-s + 5.02·8-s + 9-s − 2.45·10-s − 4.26·11-s − 4.04·12-s + 5.94·13-s − 2.32·14-s + 15-s + 4.27·16-s − 6.46·17-s + 2.45·18-s − 5.46·19-s − 4.04·20-s + 0.945·21-s − 10.4·22-s + 7.09·23-s − 5.02·24-s + 25-s + 14.6·26-s − 27-s − 3.82·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 0.577·3-s + 2.02·4-s − 0.447·5-s − 1.00·6-s − 0.357·7-s + 1.77·8-s + 0.333·9-s − 0.777·10-s − 1.28·11-s − 1.16·12-s + 1.64·13-s − 0.621·14-s + 0.258·15-s + 1.06·16-s − 1.56·17-s + 0.579·18-s − 1.25·19-s − 0.904·20-s + 0.206·21-s − 2.23·22-s + 1.48·23-s − 1.02·24-s + 0.200·25-s + 2.86·26-s − 0.192·27-s − 0.722·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 7 | \( 1 + 0.945T + 7T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 13 | \( 1 - 5.94T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 - 2.21T + 29T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 2.12T + 43T^{2} \) |
| 47 | \( 1 + 1.16T + 47T^{2} \) |
| 53 | \( 1 - 6.39T + 53T^{2} \) |
| 59 | \( 1 + 5.95T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 7.52T + 67T^{2} \) |
| 71 | \( 1 + 1.95T + 71T^{2} \) |
| 73 | \( 1 + 9.81T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 - 0.190T + 89T^{2} \) |
| 97 | \( 1 - 0.645T + 97T^{2} \) |
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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
−7.29902082413393643226962134904, −6.56436568269926915522122013702, −6.33159843850789616561855779058, −5.43985064567549910368487273462, −4.71214870065454965796240368554, −4.30440026499663950491512037980, −3.33940230640727700031077066803, −2.75928949000294855464366088989, −1.64209968997005119759118925433, 0,
1.64209968997005119759118925433, 2.75928949000294855464366088989, 3.33940230640727700031077066803, 4.30440026499663950491512037980, 4.71214870065454965796240368554, 5.43985064567549910368487273462, 6.33159843850789616561855779058, 6.56436568269926915522122013702, 7.29902082413393643226962134904
