| L(s) = 1 | − 1.12e3·2-s − 7.11e6·4-s + 4.88e7·5-s − 1.72e9·7-s + 1.74e10·8-s − 5.51e10·10-s + 1.42e12·11-s − 8.22e12·13-s + 1.94e12·14-s + 3.99e13·16-s + 5.98e12·17-s + 6.80e14·19-s − 3.47e14·20-s − 1.61e15·22-s − 1.54e13·23-s − 9.53e15·25-s + 9.27e15·26-s + 1.22e16·28-s − 1.15e17·29-s − 9.08e16·31-s − 1.91e17·32-s − 6.75e15·34-s − 8.42e16·35-s − 1.29e18·37-s − 7.67e17·38-s + 8.54e17·40-s − 5.21e18·41-s + ⋯ |
| L(s) = 1 | − 0.389·2-s − 0.848·4-s + 0.447·5-s − 0.329·7-s + 0.719·8-s − 0.174·10-s + 1.50·11-s − 1.27·13-s + 0.128·14-s + 0.567·16-s + 0.0423·17-s + 1.33·19-s − 0.379·20-s − 0.587·22-s − 0.00337·23-s − 0.799·25-s + 0.495·26-s + 0.279·28-s − 1.75·29-s − 0.642·31-s − 0.941·32-s − 0.0165·34-s − 0.147·35-s − 1.19·37-s − 0.521·38-s + 0.322·40-s − 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 141 p^{3} T + p^{23} T^{2} \) |
| 5 | \( 1 - 9772746 p T + p^{23} T^{2} \) |
| 7 | \( 1 + 35177320 p^{2} T + p^{23} T^{2} \) |
| 11 | \( 1 - 129842107284 p T + p^{23} T^{2} \) |
| 13 | \( 1 + 632381849602 p T + p^{23} T^{2} \) |
| 17 | \( 1 - 5989210330446 T + p^{23} T^{2} \) |
| 19 | \( 1 - 35789762172404 p T + p^{23} T^{2} \) |
| 23 | \( 1 + 15440648191080 T + p^{23} T^{2} \) |
| 29 | \( 1 + 115094192813324022 T + p^{23} T^{2} \) |
| 31 | \( 1 + 90829724501108800 T + p^{23} T^{2} \) |
| 37 | \( 1 + 1297873386623227570 T + p^{23} T^{2} \) |
| 41 | \( 1 + 5214036225478655130 T + p^{23} T^{2} \) |
| 43 | \( 1 + 2410434516296794108 T + p^{23} T^{2} \) |
| 47 | \( 1 - 23132669525900803824 T + p^{23} T^{2} \) |
| 53 | \( 1 - 44512631945276522850 T + p^{23} T^{2} \) |
| 59 | \( 1 - \)\(32\!\cdots\!76\)\( T + p^{23} T^{2} \) |
| 61 | \( 1 + \)\(19\!\cdots\!22\)\( T + p^{23} T^{2} \) |
| 67 | \( 1 + \)\(64\!\cdots\!96\)\( T + p^{23} T^{2} \) |
| 71 | \( 1 + \)\(35\!\cdots\!12\)\( T + p^{23} T^{2} \) |
| 73 | \( 1 - \)\(33\!\cdots\!70\)\( T + p^{23} T^{2} \) |
| 79 | \( 1 + \)\(68\!\cdots\!20\)\( T + p^{23} T^{2} \) |
| 83 | \( 1 - \)\(11\!\cdots\!44\)\( T + p^{23} T^{2} \) |
| 89 | \( 1 - \)\(23\!\cdots\!74\)\( T + p^{23} T^{2} \) |
| 97 | \( 1 + \)\(30\!\cdots\!86\)\( T + p^{23} T^{2} \) |
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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
−14.65313067847889937988967439525, −13.53937996188836486783591580294, −11.96846115743270285972359536352, −9.898686754182310474194192784888, −9.115412413380891325924885143112, −7.26655689013579548628679910478, −5.37400132264874751379354195974, −3.72936955854916153415896271450, −1.57201442821402052010783080641, 0,
1.57201442821402052010783080641, 3.72936955854916153415896271450, 5.37400132264874751379354195974, 7.26655689013579548628679910478, 9.115412413380891325924885143112, 9.898686754182310474194192784888, 11.96846115743270285972359536352, 13.53937996188836486783591580294, 14.65313067847889937988967439525
