| L(s) = 1 | + 3·3-s + 5·5-s − 12·7-s + 9·9-s − 20·11-s + 58·13-s + 15·15-s − 70·17-s − 92·19-s − 36·21-s − 112·23-s + 25·25-s + 27·27-s − 66·29-s + 108·31-s − 60·33-s − 60·35-s + 58·37-s + 174·39-s + 66·41-s − 388·43-s + 45·45-s + 408·47-s − 199·49-s − 210·51-s − 474·53-s − 100·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.647·7-s + 1/3·9-s − 0.548·11-s + 1.23·13-s + 0.258·15-s − 0.998·17-s − 1.11·19-s − 0.374·21-s − 1.01·23-s + 1/5·25-s + 0.192·27-s − 0.422·29-s + 0.625·31-s − 0.316·33-s − 0.289·35-s + 0.257·37-s + 0.714·39-s + 0.251·41-s − 1.37·43-s + 0.149·45-s + 1.26·47-s − 0.580·49-s − 0.576·51-s − 1.22·53-s − 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 58 T + p^{3} T^{2} \) |
| 41 | \( 1 - 66 T + p^{3} T^{2} \) |
| 43 | \( 1 + 388 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 474 T + p^{3} T^{2} \) |
| 59 | \( 1 + 540 T + p^{3} T^{2} \) |
| 61 | \( 1 + 14 T + p^{3} T^{2} \) |
| 67 | \( 1 + 276 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 + 790 T + p^{3} T^{2} \) |
| 79 | \( 1 + 308 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1210 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1426 T + p^{3} T^{2} \) |
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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
−9.101948771324601394897587651142, −8.543419095329913624728018495434, −7.66062542328919976649598894871, −6.45567013810365767865299374647, −6.05942045076544772532478239669, −4.66136722426304926514433948564, −3.72357113176445092896194055280, −2.67363214112413756517039431269, −1.66685661586156720958147672034, 0,
1.66685661586156720958147672034, 2.67363214112413756517039431269, 3.72357113176445092896194055280, 4.66136722426304926514433948564, 6.05942045076544772532478239669, 6.45567013810365767865299374647, 7.66062542328919976649598894871, 8.543419095329913624728018495434, 9.101948771324601394897587651142
