| L(s) = 1 | + 1.91·2-s − 3-s + 1.67·4-s + 5-s − 1.91·6-s + 2.73·7-s − 0.616·8-s + 9-s + 1.91·10-s − 3.31·11-s − 1.67·12-s + 3.77·13-s + 5.24·14-s − 15-s − 4.53·16-s − 0.701·17-s + 1.91·18-s − 5.55·19-s + 1.67·20-s − 2.73·21-s − 6.35·22-s − 7.51·23-s + 0.616·24-s + 25-s + 7.24·26-s − 27-s + 4.58·28-s + ⋯ |
| L(s) = 1 | + 1.35·2-s − 0.577·3-s + 0.839·4-s + 0.447·5-s − 0.783·6-s + 1.03·7-s − 0.217·8-s + 0.333·9-s + 0.606·10-s − 0.999·11-s − 0.484·12-s + 1.04·13-s + 1.40·14-s − 0.258·15-s − 1.13·16-s − 0.170·17-s + 0.452·18-s − 1.27·19-s + 0.375·20-s − 0.596·21-s − 1.35·22-s − 1.56·23-s + 0.125·24-s + 0.200·25-s + 1.42·26-s − 0.192·27-s + 0.867·28-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 | \( 1 - 1.91T + 2T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 3.31T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 0.701T + 17T^{2} \) |
| 19 | \( 1 + 5.55T + 19T^{2} \) |
| 23 | \( 1 + 7.51T + 23T^{2} \) |
| 29 | \( 1 + 7.35T + 29T^{2} \) |
| 31 | \( 1 + 0.768T + 31T^{2} \) |
| 37 | \( 1 + 7.01T + 37T^{2} \) |
| 41 | \( 1 - 5.50T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 6.42T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 - 7.24T + 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 - 8.44T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 - 5.97T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 9.56T + 97T^{2} \) |
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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
−7.66307302161532951073641133103, −6.58371775646290528482359944732, −6.07859184573852356910758925062, −5.46418037356865953995367964025, −4.92484549674737208514578438896, −4.18230978073073296049950254662, −3.55660723024646090308507232045, −2.31442791818475596614909836377, −1.72253885194039308858282291844, 0,
1.72253885194039308858282291844, 2.31442791818475596614909836377, 3.55660723024646090308507232045, 4.18230978073073296049950254662, 4.92484549674737208514578438896, 5.46418037356865953995367964025, 6.07859184573852356910758925062, 6.58371775646290528482359944732, 7.66307302161532951073641133103
