| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−11.84919378536414957509030361745, −10.74833060833372407783392696453, −10.00884366986370209012720211021, −8.833763178324620072523059570445, −7.53911334992942338547695306977, −6.33753201668254216191677155525, −5.41545933962092888264836465260, −3.98677613510252356626310423740, −2.14699289639219795661830303749, 0,
2.14699289639219795661830303749, 3.98677613510252356626310423740, 5.41545933962092888264836465260, 6.33753201668254216191677155525, 7.53911334992942338547695306977, 8.833763178324620072523059570445, 10.00884366986370209012720211021, 10.74833060833372407783392696453, 11.84919378536414957509030361745
