L(s) = 1 | + 2·5-s − 32·7-s + 68·11-s + 13·13-s + 14·17-s + 4·19-s − 72·23-s − 121·25-s − 102·29-s − 136·31-s − 64·35-s − 386·37-s − 250·41-s − 140·43-s + 296·47-s + 681·49-s − 526·53-s + 136·55-s − 332·59-s − 410·61-s + 26·65-s + 596·67-s + 880·71-s + 506·73-s − 2.17e3·77-s − 640·79-s − 1.38e3·83-s + ⋯ |
L(s) = 1 | + 0.178·5-s − 1.72·7-s + 1.86·11-s + 0.277·13-s + 0.199·17-s + 0.0482·19-s − 0.652·23-s − 0.967·25-s − 0.653·29-s − 0.787·31-s − 0.309·35-s − 1.71·37-s − 0.952·41-s − 0.496·43-s + 0.918·47-s + 1.98·49-s − 1.36·53-s + 0.333·55-s − 0.732·59-s − 0.860·61-s + 0.0496·65-s + 1.08·67-s + 1.47·71-s + 0.811·73-s − 3.22·77-s − 0.911·79-s − 1.82·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - p T \) |
good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 68 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 102 T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 386 T + p^{3} T^{2} \) |
| 41 | \( 1 + 250 T + p^{3} T^{2} \) |
| 43 | \( 1 + 140 T + p^{3} T^{2} \) |
| 47 | \( 1 - 296 T + p^{3} T^{2} \) |
| 53 | \( 1 + 526 T + p^{3} T^{2} \) |
| 59 | \( 1 + 332 T + p^{3} T^{2} \) |
| 61 | \( 1 + 410 T + p^{3} T^{2} \) |
| 67 | \( 1 - 596 T + p^{3} T^{2} \) |
| 71 | \( 1 - 880 T + p^{3} T^{2} \) |
| 73 | \( 1 - 506 T + p^{3} T^{2} \) |
| 79 | \( 1 + 640 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1380 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1450 T + p^{3} T^{2} \) |
| 97 | \( 1 + 446 T + p^{3} T^{2} \) |
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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
−9.854874963591984556567134082862, −9.476546513981004135339123064073, −8.579754041476501043358351941633, −7.12115902380027228718903012808, −6.45933825527282320912357798157, −5.68120483184912008031639667316, −3.96021769021079225511584685798, −3.35692263862781988931026014923, −1.67059934989288439502634804305, 0,
1.67059934989288439502634804305, 3.35692263862781988931026014923, 3.96021769021079225511584685798, 5.68120483184912008031639667316, 6.45933825527282320912357798157, 7.12115902380027228718903012808, 8.579754041476501043358351941633, 9.476546513981004135339123064073, 9.854874963591984556567134082862