| L(s) = 1 | + 3·5-s + 7-s − 4·13-s + 4·17-s − 8·19-s + 3·23-s + 4·25-s + 4·29-s + 5·31-s + 3·35-s + 11·37-s − 4·43-s + 4·47-s + 49-s + 10·53-s − 3·59-s − 12·61-s − 12·65-s − 5·67-s + 9·71-s − 16·73-s − 8·79-s − 12·83-s + 12·85-s + 13·89-s − 4·91-s − 24·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s − 1.10·13-s + 0.970·17-s − 1.83·19-s + 0.625·23-s + 4/5·25-s + 0.742·29-s + 0.898·31-s + 0.507·35-s + 1.80·37-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 1.37·53-s − 0.390·59-s − 1.53·61-s − 1.48·65-s − 0.610·67-s + 1.06·71-s − 1.87·73-s − 0.900·79-s − 1.31·83-s + 1.30·85-s + 1.37·89-s − 0.419·91-s − 2.46·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.383272463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.383272463\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−14.33401223847455, −13.78189155052504, −13.34769939304286, −12.71960037379319, −12.47944411996288, −11.74056320846945, −11.30065593395035, −10.46631963999128, −10.14869845806288, −9.932586048313433, −9.039432247140340, −8.833902750318249, −8.058965531642475, −7.539218632608408, −6.934941883424279, −6.212713607056927, −5.965432811589374, −5.305825764092852, −4.519630462721354, −4.439689039964681, −3.212918812305914, −2.605619671722271, −2.153960792537079, −1.433703240716371, −0.6264211473860857,
0.6264211473860857, 1.433703240716371, 2.153960792537079, 2.605619671722271, 3.212918812305914, 4.439689039964681, 4.519630462721354, 5.305825764092852, 5.965432811589374, 6.212713607056927, 6.934941883424279, 7.539218632608408, 8.058965531642475, 8.833902750318249, 9.039432247140340, 9.932586048313433, 10.14869845806288, 10.46631963999128, 11.30065593395035, 11.74056320846945, 12.47944411996288, 12.71960037379319, 13.34769939304286, 13.78189155052504, 14.33401223847455
