| L(s) = 1 | − 2-s + 4-s + 0.911·5-s − 4.22·7-s − 8-s − 0.911·10-s + 1.51·11-s + 0.488·13-s + 4.22·14-s + 16-s − 3.22·17-s − 5.31·19-s + 0.911·20-s − 1.51·22-s − 4.16·25-s − 0.488·26-s − 4.22·28-s + 8.00·29-s + 8.22·31-s − 32-s + 3.22·34-s − 3.85·35-s − 7.69·37-s + 5.31·38-s − 0.911·40-s + 9.19·41-s + 4.48·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.407·5-s − 1.59·7-s − 0.353·8-s − 0.288·10-s + 0.457·11-s + 0.135·13-s + 1.13·14-s + 0.250·16-s − 0.781·17-s − 1.22·19-s + 0.203·20-s − 0.323·22-s − 0.833·25-s − 0.0957·26-s − 0.799·28-s + 1.48·29-s + 1.47·31-s − 0.176·32-s + 0.552·34-s − 0.651·35-s − 1.26·37-s + 0.862·38-s − 0.144·40-s + 1.43·41-s + 0.683·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 0.911T + 5T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 - 0.488T + 13T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 + 5.31T + 19T^{2} \) |
| 29 | \( 1 - 8.00T + 29T^{2} \) |
| 31 | \( 1 - 8.22T + 31T^{2} \) |
| 37 | \( 1 + 7.69T + 37T^{2} \) |
| 41 | \( 1 - 9.19T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 - 0.273T + 47T^{2} \) |
| 53 | \( 1 - 7.33T + 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 2.04T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 0.642T + 73T^{2} \) |
| 79 | \( 1 - 1.62T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 9.35T + 89T^{2} \) |
| 97 | \( 1 - 4.56T + 97T^{2} \) |
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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
−7.21139244686309577884188222351, −6.54142677699296794615660661785, −6.34299412097670750448670825595, −5.59348626241205031742000187316, −4.39267719722108868009550979802, −3.79036637924991078163411245886, −2.75506959662460465617488258603, −2.28825750092118245877770844929, −1.03557670891665375329025100005, 0,
1.03557670891665375329025100005, 2.28825750092118245877770844929, 2.75506959662460465617488258603, 3.79036637924991078163411245886, 4.39267719722108868009550979802, 5.59348626241205031742000187316, 6.34299412097670750448670825595, 6.54142677699296794615660661785, 7.21139244686309577884188222351
