| L(s) = 1 | − 2.58·2-s − 3-s + 4.69·4-s − 5-s + 2.58·6-s + 2.48·7-s − 6.96·8-s + 9-s + 2.58·10-s − 3.95·11-s − 4.69·12-s − 1.70·13-s − 6.41·14-s + 15-s + 8.62·16-s + 1.04·17-s − 2.58·18-s + 4.06·19-s − 4.69·20-s − 2.48·21-s + 10.2·22-s − 2.67·23-s + 6.96·24-s + 25-s + 4.41·26-s − 27-s + 11.6·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.577·3-s + 2.34·4-s − 0.447·5-s + 1.05·6-s + 0.937·7-s − 2.46·8-s + 0.333·9-s + 0.817·10-s − 1.19·11-s − 1.35·12-s − 0.473·13-s − 1.71·14-s + 0.258·15-s + 2.15·16-s + 0.253·17-s − 0.609·18-s + 0.932·19-s − 1.04·20-s − 0.541·21-s + 2.18·22-s − 0.558·23-s + 1.42·24-s + 0.200·25-s + 0.865·26-s − 0.192·27-s + 2.19·28-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 | \( 1 + 2.58T + 2T^{2} \) |
| 7 | \( 1 - 2.48T + 7T^{2} \) |
| 11 | \( 1 + 3.95T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 + 5.40T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 9.41T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 + 3.53T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 5.21T + 71T^{2} \) |
| 73 | \( 1 + 5.58T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 7.23T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 3.63T + 97T^{2} \) |
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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
−7.71158439997457133972845247827, −7.53798748075497051173106115035, −6.64051937030998781286281755641, −5.62895380231335187927629371203, −5.14457666275609585764080579297, −3.99918232448062522170460363881, −2.76565750483714358541929545228, −1.98205868646783281846504596912, −0.989532561134878711020325084648, 0,
0.989532561134878711020325084648, 1.98205868646783281846504596912, 2.76565750483714358541929545228, 3.99918232448062522170460363881, 5.14457666275609585764080579297, 5.62895380231335187927629371203, 6.64051937030998781286281755641, 7.53798748075497051173106115035, 7.71158439997457133972845247827
