L(s) = 1 | − 2-s + 1.81·3-s + 4-s + 1.21·5-s − 1.81·6-s − 2.11·7-s − 8-s + 0.288·9-s − 1.21·10-s + 1.44·11-s + 1.81·12-s − 1.15·13-s + 2.11·14-s + 2.19·15-s + 16-s − 2.16·17-s − 0.288·18-s + 4.58·19-s + 1.21·20-s − 3.82·21-s − 1.44·22-s − 3.56·23-s − 1.81·24-s − 3.53·25-s + 1.15·26-s − 4.91·27-s − 2.11·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.04·3-s + 0.5·4-s + 0.541·5-s − 0.740·6-s − 0.798·7-s − 0.353·8-s + 0.0962·9-s − 0.383·10-s + 0.437·11-s + 0.523·12-s − 0.319·13-s + 0.564·14-s + 0.567·15-s + 0.250·16-s − 0.524·17-s − 0.0680·18-s + 1.05·19-s + 0.270·20-s − 0.835·21-s − 0.309·22-s − 0.742·23-s − 0.370·24-s − 0.706·25-s + 0.226·26-s − 0.946·27-s − 0.399·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 - 1.81T + 3T^{2} \) |
| 5 | \( 1 - 1.21T + 5T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 11 | \( 1 - 1.44T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 + 2.16T + 17T^{2} \) |
| 19 | \( 1 - 4.58T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 + 5.95T + 31T^{2} \) |
| 37 | \( 1 - 1.48T + 37T^{2} \) |
| 41 | \( 1 + 2.71T + 41T^{2} \) |
| 43 | \( 1 - 0.511T + 43T^{2} \) |
| 47 | \( 1 + 1.79T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 + 6.75T + 59T^{2} \) |
| 61 | \( 1 + 0.358T + 61T^{2} \) |
| 67 | \( 1 + 1.39T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 2.26T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 9.24T + 97T^{2} \) |
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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
−8.083949925666564622452432523871, −7.60691929773381507769559239548, −6.72453310074470754336650408625, −6.06528790770275680395669553983, −5.23900234466838921705715983379, −3.88673129526390236234258102628, −3.25029164417035926068921482416, −2.37924492825677895459739898670, −1.62594757864617393046661277163, 0,
1.62594757864617393046661277163, 2.37924492825677895459739898670, 3.25029164417035926068921482416, 3.88673129526390236234258102628, 5.23900234466838921705715983379, 6.06528790770275680395669553983, 6.72453310074470754336650408625, 7.60691929773381507769559239548, 8.083949925666564622452432523871