| L(s) = 1 | − 8·4-s + 5·5-s + 2·7-s + 11·11-s − 22·13-s + 64·16-s − 72·17-s + 122·19-s − 40·20-s − 72·23-s + 25·25-s − 16·28-s − 96·29-s − 112·31-s + 10·35-s + 266·37-s + 96·41-s − 382·43-s − 88·44-s − 360·47-s − 339·49-s + 176·52-s − 318·53-s + 55·55-s − 660·59-s − 430·61-s − 512·64-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.107·7-s + 0.301·11-s − 0.469·13-s + 16-s − 1.02·17-s + 1.47·19-s − 0.447·20-s − 0.652·23-s + 1/5·25-s − 0.107·28-s − 0.614·29-s − 0.648·31-s + 0.0482·35-s + 1.18·37-s + 0.365·41-s − 1.35·43-s − 0.301·44-s − 1.11·47-s − 0.988·49-s + 0.469·52-s − 0.824·53-s + 0.134·55-s − 1.45·59-s − 0.902·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 122 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 96 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 - 266 T + p^{3} T^{2} \) |
| 41 | \( 1 - 96 T + p^{3} T^{2} \) |
| 43 | \( 1 + 382 T + p^{3} T^{2} \) |
| 47 | \( 1 + 360 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 430 T + p^{3} T^{2} \) |
| 67 | \( 1 - 380 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 - 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 706 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1068 T + p^{3} T^{2} \) |
| 89 | \( 1 - 6 T + p^{3} T^{2} \) |
| 97 | \( 1 - 686 T + p^{3} T^{2} \) |
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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
−9.683102752126859252932862568205, −9.482104560567191640772841755122, −8.373739455274532644471008513846, −7.46966421242317823012824569219, −6.24540393587905710381619902510, −5.23321276756347540459846624980, −4.39569756755455242071632748698, −3.16987751593169742880091438836, −1.57594586244284102532951826945, 0,
1.57594586244284102532951826945, 3.16987751593169742880091438836, 4.39569756755455242071632748698, 5.23321276756347540459846624980, 6.24540393587905710381619902510, 7.46966421242317823012824569219, 8.373739455274532644471008513846, 9.482104560567191640772841755122, 9.683102752126859252932862568205
