| L(s) = 1 | − 4·5-s + 11-s + 2·13-s − 2·17-s − 4·19-s + 23-s + 11·25-s − 8·29-s + 4·31-s − 2·37-s − 4·41-s − 8·43-s − 8·47-s − 7·49-s − 4·53-s − 4·55-s − 4·59-s − 2·61-s − 8·65-s + 10·67-s + 6·73-s − 4·79-s − 4·83-s + 8·85-s + 16·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.208·23-s + 11/5·25-s − 1.48·29-s + 0.718·31-s − 0.328·37-s − 0.624·41-s − 1.21·43-s − 1.16·47-s − 49-s − 0.549·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s − 0.992·65-s + 1.22·67-s + 0.702·73-s − 0.450·79-s − 0.439·83-s + 0.867·85-s + 1.64·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.77755334370273, −13.30228687473297, −12.78080450099184, −12.42869125382595, −11.86983560061137, −11.37845006579406, −11.10141885678790, −10.78747135864125, −9.972012390191640, −9.535062112642312, −8.782469054577014, −8.450445456843434, −8.111680544349361, −7.568450905613502, −6.986146284077573, −6.564216325807749, −6.139624555134935, −5.158866060721922, −4.819442809302897, −4.194960756867093, −3.699416846190283, −3.388828913040949, −2.664213679089494, −1.785712251880946, −1.173261483381862, 0, 0,
1.173261483381862, 1.785712251880946, 2.664213679089494, 3.388828913040949, 3.699416846190283, 4.194960756867093, 4.819442809302897, 5.158866060721922, 6.139624555134935, 6.564216325807749, 6.986146284077573, 7.568450905613502, 8.111680544349361, 8.450445456843434, 8.782469054577014, 9.535062112642312, 9.972012390191640, 10.78747135864125, 11.10141885678790, 11.37845006579406, 11.86983560061137, 12.42869125382595, 12.78080450099184, 13.30228687473297, 13.77755334370273
