| L(s) = 1 | + 4·5-s − 4·7-s + 4·11-s − 13-s + 8·23-s + 11·25-s − 8·29-s − 4·31-s − 16·35-s + 6·37-s + 12·41-s + 8·43-s + 4·47-s + 9·49-s + 16·55-s − 4·59-s − 2·61-s − 4·65-s + 8·67-s + 4·71-s − 10·73-s − 16·77-s + 4·79-s − 12·83-s − 12·89-s + 4·91-s + 14·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.51·7-s + 1.20·11-s − 0.277·13-s + 1.66·23-s + 11/5·25-s − 1.48·29-s − 0.718·31-s − 2.70·35-s + 0.986·37-s + 1.87·41-s + 1.21·43-s + 0.583·47-s + 9/7·49-s + 2.15·55-s − 0.520·59-s − 0.256·61-s − 0.496·65-s + 0.977·67-s + 0.474·71-s − 1.17·73-s − 1.82·77-s + 0.450·79-s − 1.31·83-s − 1.27·89-s + 0.419·91-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.249762044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.249762044\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.283309900561460264303460080023, −9.043893881767278031288410702249, −7.35411058268733168359690626917, −6.72499661989682939095500288427, −5.99934179950640698199375325057, −5.54340160378585790784255817080, −4.24052892769095243824819538025, −3.14438875263608746657555973525, −2.30677622130328505633184150541, −1.07183735496432547021055620536,
1.07183735496432547021055620536, 2.30677622130328505633184150541, 3.14438875263608746657555973525, 4.24052892769095243824819538025, 5.54340160378585790784255817080, 5.99934179950640698199375325057, 6.72499661989682939095500288427, 7.35411058268733168359690626917, 9.043893881767278031288410702249, 9.283309900561460264303460080023
