L(s) = 1 | + 2.55·2-s − 0.474·3-s + 4.55·4-s − 5-s − 1.21·6-s − 7-s + 6.53·8-s − 2.77·9-s − 2.55·10-s − 0.0148·11-s − 2.16·12-s + 1.86·13-s − 2.55·14-s + 0.474·15-s + 7.62·16-s − 5.80·17-s − 7.10·18-s + 0.184·19-s − 4.55·20-s + 0.474·21-s − 0.0380·22-s + 8.06·23-s − 3.10·24-s + 25-s + 4.77·26-s + 2.74·27-s − 4.55·28-s + ⋯ |
L(s) = 1 | + 1.81·2-s − 0.274·3-s + 2.27·4-s − 0.447·5-s − 0.496·6-s − 0.377·7-s + 2.31·8-s − 0.924·9-s − 0.809·10-s − 0.00448·11-s − 0.623·12-s + 0.517·13-s − 0.684·14-s + 0.122·15-s + 1.90·16-s − 1.40·17-s − 1.67·18-s + 0.0422·19-s − 1.01·20-s + 0.103·21-s − 0.00811·22-s + 1.68·23-s − 0.633·24-s + 0.200·25-s + 0.936·26-s + 0.527·27-s − 0.860·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 3 | \( 1 + 0.474T + 3T^{2} \) |
| 11 | \( 1 + 0.0148T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 - 0.184T + 19T^{2} \) |
| 23 | \( 1 - 8.06T + 23T^{2} \) |
| 29 | \( 1 + 8.64T + 29T^{2} \) |
| 31 | \( 1 + 8.09T + 31T^{2} \) |
| 37 | \( 1 + 0.537T + 37T^{2} \) |
| 41 | \( 1 - 3.28T + 41T^{2} \) |
| 43 | \( 1 - 8.52T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 0.140T + 53T^{2} \) |
| 59 | \( 1 - 0.975T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 + 9.57T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 2.27T + 73T^{2} \) |
| 79 | \( 1 - 3.35T + 79T^{2} \) |
| 83 | \( 1 + 6.73T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 6.50T + 97T^{2} \) |
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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
−7.07007730224209401314163983589, −6.66990997462402173241853188164, −5.80046697535991570464406134559, −5.47400987992272476234483279149, −4.60363974133426549640430480560, −4.00647457088418023795087296738, −3.22328501775258263232836678594, −2.70640936426436890175719045357, −1.64869683183021932637729036347, 0,
1.64869683183021932637729036347, 2.70640936426436890175719045357, 3.22328501775258263232836678594, 4.00647457088418023795087296738, 4.60363974133426549640430480560, 5.47400987992272476234483279149, 5.80046697535991570464406134559, 6.66990997462402173241853188164, 7.07007730224209401314163983589