L(s) = 1 | + 6.44·2-s + 7.63·3-s + 9.56·4-s + 25·5-s + 49.2·6-s + 186.·7-s − 144.·8-s − 184.·9-s + 161.·10-s + 121·11-s + 72.9·12-s − 69.3·13-s + 1.20e3·14-s + 190.·15-s − 1.23e3·16-s − 662.·17-s − 1.19e3·18-s − 361·19-s + 239.·20-s + 1.42e3·21-s + 780.·22-s − 1.30e3·23-s − 1.10e3·24-s + 625·25-s − 446.·26-s − 3.26e3·27-s + 1.78e3·28-s + ⋯ |
L(s) = 1 | + 1.13·2-s + 0.489·3-s + 0.298·4-s + 0.447·5-s + 0.558·6-s + 1.43·7-s − 0.799·8-s − 0.760·9-s + 0.509·10-s + 0.301·11-s + 0.146·12-s − 0.113·13-s + 1.63·14-s + 0.218·15-s − 1.20·16-s − 0.555·17-s − 0.866·18-s − 0.229·19-s + 0.133·20-s + 0.703·21-s + 0.343·22-s − 0.514·23-s − 0.391·24-s + 0.200·25-s − 0.129·26-s − 0.861·27-s + 0.429·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
good | 2 | \( 1 - 6.44T + 32T^{2} \) |
| 3 | \( 1 - 7.63T + 243T^{2} \) |
| 7 | \( 1 - 186.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 69.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 662.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.60e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.76e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.16e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.67e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.24e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.00e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.78e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.32e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.19e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.44e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.54e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.51e5T + 8.58e9T^{2} \) |
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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
−8.746326441230840975923549557217, −8.062440991649891215788104537550, −6.94629527934072494216702571293, −5.85339399775043641366629769127, −5.27014747041787527924122991419, −4.42373844748455605088897525239, −3.56748865024448102994710245467, −2.46683652396760620881389589545, −1.68166238640961049756371469717, 0,
1.68166238640961049756371469717, 2.46683652396760620881389589545, 3.56748865024448102994710245467, 4.42373844748455605088897525239, 5.27014747041787527924122991419, 5.85339399775043641366629769127, 6.94629527934072494216702571293, 8.062440991649891215788104537550, 8.746326441230840975923549557217