| L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s + 4·7-s − 8-s + 6·9-s + 3·11-s + 3·12-s − 6·13-s − 4·14-s + 16-s + 5·17-s − 6·18-s − 19-s + 12·21-s − 3·22-s + 23-s − 3·24-s + 6·26-s + 9·27-s + 4·28-s − 8·29-s − 8·31-s − 32-s + 9·33-s − 5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s + 1.51·7-s − 0.353·8-s + 2·9-s + 0.904·11-s + 0.866·12-s − 1.66·13-s − 1.06·14-s + 1/4·16-s + 1.21·17-s − 1.41·18-s − 0.229·19-s + 2.61·21-s − 0.639·22-s + 0.208·23-s − 0.612·24-s + 1.17·26-s + 1.73·27-s + 0.755·28-s − 1.48·29-s − 1.43·31-s − 0.176·32-s + 1.56·33-s − 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.562205493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.562205493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−9.418697221621826698091322517723, −9.085426534280127450003684766543, −8.117541525349201165845174784319, −7.59474610161940484578941715279, −7.12956679221094430486867721686, −5.46616785894646380813647164423, −4.38590550697342217753796693067, −3.37416141049219083022255530454, −2.20877669468819065006590445959, −1.52598777083604923783702801600,
1.52598777083604923783702801600, 2.20877669468819065006590445959, 3.37416141049219083022255530454, 4.38590550697342217753796693067, 5.46616785894646380813647164423, 7.12956679221094430486867721686, 7.59474610161940484578941715279, 8.117541525349201165845174784319, 9.085426534280127450003684766543, 9.418697221621826698091322517723
