| L(s) = 1 | + 6·5-s − 7·7-s − 30·11-s + 2·13-s + 66·17-s + 52·19-s − 114·23-s − 89·25-s + 72·29-s + 196·31-s − 42·35-s − 286·37-s − 378·41-s − 164·43-s + 228·47-s + 49·49-s − 348·53-s − 180·55-s + 348·59-s − 106·61-s + 12·65-s − 596·67-s − 630·71-s − 1.04e3·73-s + 210·77-s + 88·79-s + 1.44e3·83-s + ⋯ |
| L(s) = 1 | + 0.536·5-s − 0.377·7-s − 0.822·11-s + 0.0426·13-s + 0.941·17-s + 0.627·19-s − 1.03·23-s − 0.711·25-s + 0.461·29-s + 1.13·31-s − 0.202·35-s − 1.27·37-s − 1.43·41-s − 0.581·43-s + 0.707·47-s + 1/7·49-s − 0.901·53-s − 0.441·55-s + 0.767·59-s − 0.222·61-s + 0.0228·65-s − 1.08·67-s − 1.05·71-s − 1.67·73-s + 0.310·77-s + 0.125·79-s + 1.90·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 114 T + p^{3} T^{2} \) |
| 29 | \( 1 - 72 T + p^{3} T^{2} \) |
| 31 | \( 1 - 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 228 T + p^{3} T^{2} \) |
| 53 | \( 1 + 348 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 + 106 T + p^{3} T^{2} \) |
| 67 | \( 1 + 596 T + p^{3} T^{2} \) |
| 71 | \( 1 + 630 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1042 T + p^{3} T^{2} \) |
| 79 | \( 1 - 88 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1440 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1374 T + p^{3} T^{2} \) |
| 97 | \( 1 + 34 T + p^{3} T^{2} \) |
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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
−9.286677259423209856142800999929, −8.265619011334003240264128245354, −7.58211309434249137041517064363, −6.51930321878105837590472212703, −5.70998616789930556889720518778, −4.94403847263688634717695576061, −3.64180893057964741937494503871, −2.69643075562162236542044907966, −1.49307987765750710332410259737, 0,
1.49307987765750710332410259737, 2.69643075562162236542044907966, 3.64180893057964741937494503871, 4.94403847263688634717695576061, 5.70998616789930556889720518778, 6.51930321878105837590472212703, 7.58211309434249137041517064363, 8.265619011334003240264128245354, 9.286677259423209856142800999929
