| L(s) = 1 | + 4·2-s + 16·4-s − 26·5-s − 49·7-s + 64·8-s − 104·10-s − 664·11-s + 318·13-s − 196·14-s + 256·16-s − 1.58e3·17-s + 236·19-s − 416·20-s − 2.65e3·22-s − 2.21e3·23-s − 2.44e3·25-s + 1.27e3·26-s − 784·28-s + 4.95e3·29-s − 7.12e3·31-s + 1.02e3·32-s − 6.32e3·34-s + 1.27e3·35-s + 4.35e3·37-s + 944·38-s − 1.66e3·40-s − 1.05e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.465·5-s − 0.377·7-s + 0.353·8-s − 0.328·10-s − 1.65·11-s + 0.521·13-s − 0.267·14-s + 1/4·16-s − 1.32·17-s + 0.149·19-s − 0.232·20-s − 1.16·22-s − 0.871·23-s − 0.783·25-s + 0.369·26-s − 0.188·28-s + 1.09·29-s − 1.33·31-s + 0.176·32-s − 0.938·34-s + 0.175·35-s + 0.523·37-s + 0.106·38-s − 0.164·40-s − 0.979·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 26 T + p^{5} T^{2} \) |
| 11 | \( 1 + 664 T + p^{5} T^{2} \) |
| 13 | \( 1 - 318 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1582 T + p^{5} T^{2} \) |
| 19 | \( 1 - 236 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2212 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4954 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7128 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4358 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10542 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8452 T + p^{5} T^{2} \) |
| 47 | \( 1 + 5352 T + p^{5} T^{2} \) |
| 53 | \( 1 - 33354 T + p^{5} T^{2} \) |
| 59 | \( 1 - 15436 T + p^{5} T^{2} \) |
| 61 | \( 1 + 36762 T + p^{5} T^{2} \) |
| 67 | \( 1 - 40972 T + p^{5} T^{2} \) |
| 71 | \( 1 - 9092 T + p^{5} T^{2} \) |
| 73 | \( 1 + 73454 T + p^{5} T^{2} \) |
| 79 | \( 1 - 89400 T + p^{5} T^{2} \) |
| 83 | \( 1 - 6428 T + p^{5} T^{2} \) |
| 89 | \( 1 - 122658 T + p^{5} T^{2} \) |
| 97 | \( 1 - 21370 T + p^{5} T^{2} \) |
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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
−12.04022695383416727624633704007, −11.01801208592486658423154011514, −10.10718423441914842894614593599, −8.523281763303461303593618915159, −7.47466317843497847793844847019, −6.21555441545313718617996499970, −4.98936430784473611144674787749, −3.69516476894819029975161239788, −2.31485346858646908750879863284, 0,
2.31485346858646908750879863284, 3.69516476894819029975161239788, 4.98936430784473611144674787749, 6.21555441545313718617996499970, 7.47466317843497847793844847019, 8.523281763303461303593618915159, 10.10718423441914842894614593599, 11.01801208592486658423154011514, 12.04022695383416727624633704007
