| L(s) = 1 | + 5·5-s − 20·7-s − 56·11-s − 86·13-s + 106·17-s − 4·19-s + 136·23-s + 25·25-s + 206·29-s + 152·31-s − 100·35-s + 282·37-s + 246·41-s − 412·43-s + 40·47-s + 57·49-s + 126·53-s − 280·55-s + 56·59-s − 2·61-s − 430·65-s + 388·67-s − 672·71-s + 1.17e3·73-s + 1.12e3·77-s − 408·79-s + 668·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.07·7-s − 1.53·11-s − 1.83·13-s + 1.51·17-s − 0.0482·19-s + 1.23·23-s + 1/5·25-s + 1.31·29-s + 0.880·31-s − 0.482·35-s + 1.25·37-s + 0.937·41-s − 1.46·43-s + 0.124·47-s + 0.166·49-s + 0.326·53-s − 0.686·55-s + 0.123·59-s − 0.00419·61-s − 0.820·65-s + 0.707·67-s − 1.12·71-s + 1.87·73-s + 1.65·77-s − 0.581·79-s + 0.883·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.445930648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.445930648\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 56 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 - 106 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 136 T + p^{3} T^{2} \) |
| 29 | \( 1 - 206 T + p^{3} T^{2} \) |
| 31 | \( 1 - 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 282 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 40 T + p^{3} T^{2} \) |
| 53 | \( 1 - 126 T + p^{3} T^{2} \) |
| 59 | \( 1 - 56 T + p^{3} T^{2} \) |
| 61 | \( 1 + 2 T + p^{3} T^{2} \) |
| 67 | \( 1 - 388 T + p^{3} T^{2} \) |
| 71 | \( 1 + 672 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1170 T + p^{3} T^{2} \) |
| 79 | \( 1 + 408 T + p^{3} T^{2} \) |
| 83 | \( 1 - 668 T + p^{3} T^{2} \) |
| 89 | \( 1 + 66 T + p^{3} T^{2} \) |
| 97 | \( 1 + 926 T + p^{3} T^{2} \) |
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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
−9.966096515770255838195583686179, −9.493864387188800645835462542262, −8.166099746388934360114435975072, −7.41528957462770992748127007713, −6.50942037724942214409297562249, −5.42853774452513999255338041522, −4.77375497356854754717661725198, −3.05806569117810373200245554217, −2.56337424129377760391774000581, −0.66180652198307057085647703255,
0.66180652198307057085647703255, 2.56337424129377760391774000581, 3.05806569117810373200245554217, 4.77375497356854754717661725198, 5.42853774452513999255338041522, 6.50942037724942214409297562249, 7.41528957462770992748127007713, 8.166099746388934360114435975072, 9.493864387188800645835462542262, 9.966096515770255838195583686179
