| L(s) = 1 | + 3·5-s − 7-s + 3·11-s − 4·13-s + 6·17-s + 7·19-s − 3·23-s + 4·25-s − 5·31-s − 3·35-s − 7·37-s + 9·41-s + 10·43-s + 6·47-s + 49-s − 12·53-s + 9·55-s − 6·59-s + 8·61-s − 12·65-s + 4·67-s + 9·71-s + 2·73-s − 3·77-s + 10·79-s + 18·85-s − 15·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 0.904·11-s − 1.10·13-s + 1.45·17-s + 1.60·19-s − 0.625·23-s + 4/5·25-s − 0.898·31-s − 0.507·35-s − 1.15·37-s + 1.40·41-s + 1.52·43-s + 0.875·47-s + 1/7·49-s − 1.64·53-s + 1.21·55-s − 0.781·59-s + 1.02·61-s − 1.48·65-s + 0.488·67-s + 1.06·71-s + 0.234·73-s − 0.341·77-s + 1.12·79-s + 1.95·85-s − 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.524412417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.524412417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.078219390102068883937743483807, −7.78147488690010910533658209444, −7.27001733508098481035370711240, −6.33551791202706693378801482159, −5.63118792961852328590051410411, −5.15731301288190726491879543639, −3.89997334478161229253100325219, −3.00878163634244191829198143524, −2.04047967619860809562977845320, −1.02598251138164473780947138949,
1.02598251138164473780947138949, 2.04047967619860809562977845320, 3.00878163634244191829198143524, 3.89997334478161229253100325219, 5.15731301288190726491879543639, 5.63118792961852328590051410411, 6.33551791202706693378801482159, 7.27001733508098481035370711240, 7.78147488690010910533658209444, 9.078219390102068883937743483807
