| 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
−9.165768985182441092986881710187, −8.669970855168011631271862741698, −7.31319973861422706965103338132, −6.37542473637401147772015462070, −6.12537514395331728098940974280, −4.65733312212444274805595891188, −4.08976922779465699752526250589, −2.55176088597057365657437393161, −1.41961697828443694613888966486, 0,
1.41961697828443694613888966486, 2.55176088597057365657437393161, 4.08976922779465699752526250589, 4.65733312212444274805595891188, 6.12537514395331728098940974280, 6.37542473637401147772015462070, 7.31319973861422706965103338132, 8.669970855168011631271862741698, 9.165768985182441092986881710187
