| L(s) = 1 | + 2·5-s + 7-s − 3·9-s + 11-s + 2·13-s + 2·17-s + 8·23-s − 25-s − 2·29-s + 8·31-s + 2·35-s − 2·37-s + 10·41-s − 4·43-s − 6·45-s − 8·47-s + 49-s + 6·53-s + 2·55-s + 10·61-s − 3·63-s + 4·65-s + 12·67-s − 16·71-s − 14·73-s + 77-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 9-s + 0.301·11-s + 0.554·13-s + 0.485·17-s + 1.66·23-s − 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s + 1.56·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.269·55-s + 1.28·61-s − 0.377·63-s + 0.496·65-s + 1.46·67-s − 1.89·71-s − 1.63·73-s + 0.113·77-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.026025029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026025029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.677111884773371688797795403536, −8.873654783935412415224532648296, −8.285581166378550997277831793349, −7.20326435621310186464970887967, −6.21550319266596796324909974948, −5.60979046831661786876827516293, −4.72881899635054621686684497745, −3.41143066403285592604947569117, −2.43425506739898130663595022887, −1.14548816396094190613141167258,
1.14548816396094190613141167258, 2.43425506739898130663595022887, 3.41143066403285592604947569117, 4.72881899635054621686684497745, 5.60979046831661786876827516293, 6.21550319266596796324909974948, 7.20326435621310186464970887967, 8.285581166378550997277831793349, 8.873654783935412415224532648296, 9.677111884773371688797795403536
