| L(s) = 1 | − 1.54·2-s + 1.51·3-s + 0.390·4-s + 5-s − 2.33·6-s − 4.16·7-s + 2.48·8-s − 0.709·9-s − 1.54·10-s − 11-s + 0.591·12-s + 1.82·13-s + 6.43·14-s + 1.51·15-s − 4.62·16-s + 4.80·17-s + 1.09·18-s + 19-s + 0.390·20-s − 6.29·21-s + 1.54·22-s − 5.53·23-s + 3.76·24-s + 25-s − 2.82·26-s − 5.61·27-s − 1.62·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.873·3-s + 0.195·4-s + 0.447·5-s − 0.955·6-s − 1.57·7-s + 0.879·8-s − 0.236·9-s − 0.488·10-s − 0.301·11-s + 0.170·12-s + 0.507·13-s + 1.71·14-s + 0.390·15-s − 1.15·16-s + 1.16·17-s + 0.258·18-s + 0.229·19-s + 0.0873·20-s − 1.37·21-s + 0.329·22-s − 1.15·23-s + 0.768·24-s + 0.200·25-s − 0.554·26-s − 1.08·27-s − 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.54T + 2T^{2} \) |
| 3 | \( 1 - 1.51T + 3T^{2} \) |
| 7 | \( 1 + 4.16T + 7T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 23 | \( 1 + 5.53T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 + 7.11T + 37T^{2} \) |
| 41 | \( 1 + 3.85T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 5.75T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 7.35T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 4.55T + 71T^{2} \) |
| 73 | \( 1 + 6.03T + 73T^{2} \) |
| 79 | \( 1 - 9.68T + 79T^{2} \) |
| 83 | \( 1 - 0.508T + 83T^{2} \) |
| 89 | \( 1 - 6.86T + 89T^{2} \) |
| 97 | \( 1 + 3.72T + 97T^{2} \) |
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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
−9.410199096622593530288356575707, −8.840940622397730716856167431817, −8.094304179704830864564597772069, −7.28976318400487319468389766219, −6.28881268833515668189081762988, −5.35707799224462277849597441460, −3.74060626740907007332681735399, −3.03566775091113100305494993993, −1.72958771278925189564283111174, 0,
1.72958771278925189564283111174, 3.03566775091113100305494993993, 3.74060626740907007332681735399, 5.35707799224462277849597441460, 6.28881268833515668189081762988, 7.28976318400487319468389766219, 8.094304179704830864564597772069, 8.840940622397730716856167431817, 9.410199096622593530288356575707
