| L(s) = 1 | + 4·7-s + 2·11-s − 4·13-s − 6·17-s + 4·19-s + 23-s − 8·29-s − 8·31-s + 10·37-s − 6·41-s + 6·43-s + 4·47-s + 9·49-s − 14·53-s + 4·59-s + 6·61-s + 14·67-s + 10·71-s − 14·73-s + 8·77-s + 8·79-s + 4·83-s − 16·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.603·11-s − 1.10·13-s − 1.45·17-s + 0.917·19-s + 0.208·23-s − 1.48·29-s − 1.43·31-s + 1.64·37-s − 0.937·41-s + 0.914·43-s + 0.583·47-s + 9/7·49-s − 1.92·53-s + 0.520·59-s + 0.768·61-s + 1.71·67-s + 1.18·71-s − 1.63·73-s + 0.911·77-s + 0.900·79-s + 0.439·83-s − 1.67·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.499024431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.499024431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.20042021168770, −13.38136484596899, −13.05526948003818, −12.44093352316385, −11.85673061872930, −11.40303128702121, −11.04391671831568, −10.75967169231850, −9.770559469481716, −9.390309410794759, −9.056373744836454, −8.320988991032722, −7.834147231164858, −7.363620152758068, −6.952877066410577, −6.247304936566403, −5.471439322340474, −5.137402097848547, −4.545398001761438, −4.066542939702825, −3.406821266030156, −2.420653747668682, −2.050346112076994, −1.404221711538046, −0.5093760678740160,
0.5093760678740160, 1.404221711538046, 2.050346112076994, 2.420653747668682, 3.406821266030156, 4.066542939702825, 4.545398001761438, 5.137402097848547, 5.471439322340474, 6.247304936566403, 6.952877066410577, 7.363620152758068, 7.834147231164858, 8.320988991032722, 9.056373744836454, 9.390309410794759, 9.770559469481716, 10.75967169231850, 11.04391671831568, 11.40303128702121, 11.85673061872930, 12.44093352316385, 13.05526948003818, 13.38136484596899, 14.20042021168770
