| L(s) = 1 | − 1.77e5·3-s − 4.88e7·5-s + 1.72e9·7-s + 3.13e10·9-s + 1.42e12·11-s − 8.22e12·13-s + 8.65e12·15-s − 5.98e12·17-s − 6.80e14·19-s − 3.05e14·21-s − 1.54e13·23-s − 9.53e15·25-s − 5.55e15·27-s + 1.15e17·29-s + 9.08e16·31-s − 2.53e17·33-s − 8.42e16·35-s − 1.29e18·37-s + 1.45e18·39-s + 5.21e18·41-s + 2.41e18·43-s − 1.53e18·45-s + 2.31e19·47-s − 2.43e19·49-s + 1.06e18·51-s − 4.45e19·53-s − 6.97e19·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.329·7-s + 1/3·9-s + 1.50·11-s − 1.27·13-s + 0.258·15-s − 0.0423·17-s − 1.33·19-s − 0.190·21-s − 0.00337·23-s − 0.799·25-s − 0.192·27-s + 1.75·29-s + 0.642·31-s − 0.871·33-s − 0.147·35-s − 1.19·37-s + 0.734·39-s + 1.47·41-s + 0.395·43-s − 0.149·45-s + 1.36·47-s − 0.891·49-s + 0.0244·51-s − 0.659·53-s − 0.675·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + p^{11} T \) |
| good | 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
−10.75967858661460162090564934586, −9.593801703422847802782498658088, −8.365352403386498479799244708784, −7.09990119689153597539638690338, −6.18299981059104311479472306819, −4.73804068996480651754075944873, −3.98071394945233049586377912456, −2.35764717769058245240720713761, −1.10700616729042341780425901424, 0,
1.10700616729042341780425901424, 2.35764717769058245240720713761, 3.98071394945233049586377912456, 4.73804068996480651754075944873, 6.18299981059104311479472306819, 7.09990119689153597539638690338, 8.365352403386498479799244708784, 9.593801703422847802782498658088, 10.75967858661460162090564934586
