| L(s) = 1 | − 2.06·2-s − 3-s + 2.25·4-s − 2.15·5-s + 2.06·6-s + 0.293·7-s − 0.535·8-s + 9-s + 4.45·10-s − 1.28·11-s − 2.25·12-s + 6.19·13-s − 0.604·14-s + 2.15·15-s − 3.41·16-s + 17-s − 2.06·18-s − 7.51·19-s − 4.87·20-s − 0.293·21-s + 2.66·22-s + 8.06·23-s + 0.535·24-s − 0.340·25-s − 12.7·26-s − 27-s + 0.662·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 0.577·3-s + 1.12·4-s − 0.965·5-s + 0.842·6-s + 0.110·7-s − 0.189·8-s + 0.333·9-s + 1.40·10-s − 0.388·11-s − 0.652·12-s + 1.71·13-s − 0.161·14-s + 0.557·15-s − 0.853·16-s + 0.242·17-s − 0.486·18-s − 1.72·19-s − 1.09·20-s − 0.0639·21-s + 0.567·22-s + 1.68·23-s + 0.109·24-s − 0.0681·25-s − 2.50·26-s − 0.192·27-s + 0.125·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 5 | \( 1 + 2.15T + 5T^{2} \) |
| 7 | \( 1 - 0.293T + 7T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 - 6.19T + 13T^{2} \) |
| 19 | \( 1 + 7.51T + 19T^{2} \) |
| 23 | \( 1 - 8.06T + 23T^{2} \) |
| 29 | \( 1 + 2.92T + 29T^{2} \) |
| 31 | \( 1 + 0.441T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 - 7.73T + 41T^{2} \) |
| 43 | \( 1 + 9.62T + 43T^{2} \) |
| 47 | \( 1 - 0.0536T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.28T + 59T^{2} \) |
| 61 | \( 1 + 3.62T + 61T^{2} \) |
| 67 | \( 1 + 7.01T + 67T^{2} \) |
| 71 | \( 1 + 7.52T + 71T^{2} \) |
| 73 | \( 1 + 4.73T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 4.35T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.65588089559112425018634687595, −6.99880801800159503376756009355, −6.36619390562644874206893157145, −5.59589001107362055061048958879, −4.52005055181421514483121272695, −4.00591731299221084467695443636, −2.97321841030016290202410703085, −1.76848236470033838853502250744, −0.935124855606542175223780317172, 0,
0.935124855606542175223780317172, 1.76848236470033838853502250744, 2.97321841030016290202410703085, 4.00591731299221084467695443636, 4.52005055181421514483121272695, 5.59589001107362055061048958879, 6.36619390562644874206893157145, 6.99880801800159503376756009355, 7.65588089559112425018634687595
