| L(s) = 1 | + 1.12e3·2-s + 1.77e5·3-s − 7.11e6·4-s − 4.88e7·5-s + 1.99e8·6-s − 1.72e9·7-s − 1.74e10·8-s + 3.13e10·9-s − 5.51e10·10-s − 1.42e12·11-s − 1.26e12·12-s − 8.22e12·13-s − 1.94e12·14-s − 8.65e12·15-s + 3.99e13·16-s − 5.98e12·17-s + 3.53e13·18-s + 6.80e14·19-s + 3.47e14·20-s − 3.05e14·21-s − 1.61e15·22-s + 1.54e13·23-s − 3.09e15·24-s − 9.53e15·25-s − 9.27e15·26-s + 5.55e15·27-s + 1.22e16·28-s + ⋯ |
| L(s) = 1 | + 0.389·2-s + 0.577·3-s − 0.848·4-s − 0.447·5-s + 0.224·6-s − 0.329·7-s − 0.719·8-s + 1/3·9-s − 0.174·10-s − 1.50·11-s − 0.489·12-s − 1.27·13-s − 0.128·14-s − 0.258·15-s + 0.567·16-s − 0.0423·17-s + 0.129·18-s + 1.33·19-s + 0.379·20-s − 0.190·21-s − 0.587·22-s + 0.00337·23-s − 0.415·24-s − 0.799·25-s − 0.495·26-s + 0.192·27-s + 0.279·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{11} T \) |
| 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
−19.55848746890403235781033512048, −18.03792405101557126482274858790, −15.65349536356501171189382382218, −14.00196899466533255903183083900, −12.53013863163623625579100748314, −9.770527041793505405321287691707, −7.85748098486953502191745876535, −4.95545867931327919581802415184, −3.04577515234153929617839320136, 0,
3.04577515234153929617839320136, 4.95545867931327919581802415184, 7.85748098486953502191745876535, 9.770527041793505405321287691707, 12.53013863163623625579100748314, 14.00196899466533255903183083900, 15.65349536356501171189382382218, 18.03792405101557126482274858790, 19.55848746890403235781033512048
