| L(s) = 1 | + 2·5-s + 2·7-s + 11-s − 2·13-s + 2·19-s − 25-s + 8·29-s + 4·31-s + 4·35-s − 6·37-s + 4·41-s + 6·43-s + 8·47-s − 3·49-s + 6·53-s + 2·55-s + 4·59-s − 6·61-s − 4·65-s + 4·67-s + 8·71-s − 10·73-s + 2·77-s − 14·79-s + 8·83-s − 2·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.301·11-s − 0.554·13-s + 0.458·19-s − 1/5·25-s + 1.48·29-s + 0.718·31-s + 0.676·35-s − 0.986·37-s + 0.624·41-s + 0.914·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.269·55-s + 0.520·59-s − 0.768·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.227·77-s − 1.57·79-s + 0.878·83-s − 0.211·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.545061417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.545061417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.741432993876465800467764479640, −7.937839645410758533791863487956, −7.19306300393180880258724467565, −6.36356837214073433395665681088, −5.61034403221069151944874816863, −4.90005158239302952603845530456, −4.11178662790885532582576400158, −2.86651983749511535818482386419, −2.04996257344413336974576420239, −1.01951695186635853569797311441,
1.01951695186635853569797311441, 2.04996257344413336974576420239, 2.86651983749511535818482386419, 4.11178662790885532582576400158, 4.90005158239302952603845530456, 5.61034403221069151944874816863, 6.36356837214073433395665681088, 7.19306300393180880258724467565, 7.937839645410758533791863487956, 8.741432993876465800467764479640
