| L(s) = 1 | + 3·3-s − 4·5-s − 7·7-s + 9·9-s + 26·11-s − 2·13-s − 12·15-s − 36·17-s + 76·19-s − 21·21-s − 114·23-s − 109·25-s + 27·27-s − 6·29-s − 256·31-s + 78·33-s + 28·35-s + 86·37-s − 6·39-s + 160·41-s + 220·43-s − 36·45-s + 308·47-s + 49·49-s − 108·51-s − 258·53-s − 104·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.357·5-s − 0.377·7-s + 1/3·9-s + 0.712·11-s − 0.0426·13-s − 0.206·15-s − 0.513·17-s + 0.917·19-s − 0.218·21-s − 1.03·23-s − 0.871·25-s + 0.192·27-s − 0.0384·29-s − 1.48·31-s + 0.411·33-s + 0.135·35-s + 0.382·37-s − 0.0246·39-s + 0.609·41-s + 0.780·43-s − 0.119·45-s + 0.955·47-s + 1/7·49-s − 0.296·51-s − 0.668·53-s − 0.254·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 26 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 36 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 114 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 256 T + p^{3} T^{2} \) |
| 37 | \( 1 - 86 T + p^{3} T^{2} \) |
| 41 | \( 1 - 160 T + p^{3} T^{2} \) |
| 43 | \( 1 - 220 T + p^{3} T^{2} \) |
| 47 | \( 1 - 308 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 + 264 T + p^{3} T^{2} \) |
| 61 | \( 1 + 606 T + p^{3} T^{2} \) |
| 67 | \( 1 - 520 T + p^{3} T^{2} \) |
| 71 | \( 1 + 286 T + p^{3} T^{2} \) |
| 73 | \( 1 + 530 T + p^{3} T^{2} \) |
| 79 | \( 1 + 44 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1012 T + p^{3} T^{2} \) |
| 89 | \( 1 - 768 T + p^{3} T^{2} \) |
| 97 | \( 1 - 222 T + p^{3} T^{2} \) |
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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
−9.064530823281896711467376227918, −7.896646344593523651390438319743, −7.41055323808442967066688452587, −6.41033146099019580674548985826, −5.56101660103555262183897073934, −4.25334098512350322093309588906, −3.69291039263940422712898344595, −2.59825537064441812945556532234, −1.45528507701977829944091918124, 0,
1.45528507701977829944091918124, 2.59825537064441812945556532234, 3.69291039263940422712898344595, 4.25334098512350322093309588906, 5.56101660103555262183897073934, 6.41033146099019580674548985826, 7.41055323808442967066688452587, 7.896646344593523651390438319743, 9.064530823281896711467376227918
