| L(s) = 1 | + 3·3-s + 5·5-s + 4·7-s + 9·9-s − 48·11-s − 2·13-s + 15·15-s − 114·17-s + 140·19-s + 12·21-s − 72·23-s + 25·25-s + 27·27-s − 210·29-s − 272·31-s − 144·33-s + 20·35-s + 334·37-s − 6·39-s − 198·41-s − 268·43-s + 45·45-s − 216·47-s − 327·49-s − 342·51-s + 78·53-s − 240·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.215·7-s + 1/3·9-s − 1.31·11-s − 0.0426·13-s + 0.258·15-s − 1.62·17-s + 1.69·19-s + 0.124·21-s − 0.652·23-s + 1/5·25-s + 0.192·27-s − 1.34·29-s − 1.57·31-s − 0.759·33-s + 0.0965·35-s + 1.48·37-s − 0.0246·39-s − 0.754·41-s − 0.950·43-s + 0.149·45-s − 0.670·47-s − 0.953·49-s − 0.939·51-s + 0.202·53-s − 0.588·55-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 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 140 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 210 T + p^{3} T^{2} \) |
| 31 | \( 1 + 272 T + p^{3} T^{2} \) |
| 37 | \( 1 - 334 T + p^{3} T^{2} \) |
| 41 | \( 1 + 198 T + p^{3} T^{2} \) |
| 43 | \( 1 + 268 T + p^{3} T^{2} \) |
| 47 | \( 1 + 216 T + p^{3} T^{2} \) |
| 53 | \( 1 - 78 T + p^{3} T^{2} \) |
| 59 | \( 1 - 240 T + p^{3} T^{2} \) |
| 61 | \( 1 + 302 T + p^{3} T^{2} \) |
| 67 | \( 1 - 596 T + p^{3} T^{2} \) |
| 71 | \( 1 - 768 T + p^{3} T^{2} \) |
| 73 | \( 1 + 478 T + p^{3} T^{2} \) |
| 79 | \( 1 - 640 T + p^{3} T^{2} \) |
| 83 | \( 1 + 348 T + p^{3} T^{2} \) |
| 89 | \( 1 - 210 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1534 T + p^{3} T^{2} \) |
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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
−9.373897203801679270685986440114, −8.326636274833327531021513804600, −7.65386536991229911023374386101, −6.80988309306848538105040449994, −5.60620648898840798030299316611, −4.91121497346596219161785040443, −3.68335909929705216953658165337, −2.59957517821615905574863321613, −1.72593322052405312703399137589, 0,
1.72593322052405312703399137589, 2.59957517821615905574863321613, 3.68335909929705216953658165337, 4.91121497346596219161785040443, 5.60620648898840798030299316611, 6.80988309306848538105040449994, 7.65386536991229911023374386101, 8.326636274833327531021513804600, 9.373897203801679270685986440114
