| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s − 4·11-s + 2·13-s + 15-s + 2·17-s + 19-s − 4·21-s + 4·23-s + 25-s + 27-s − 2·29-s − 4·33-s − 4·35-s − 6·37-s + 2·39-s − 6·41-s − 8·43-s + 45-s + 12·47-s + 9·49-s + 2·51-s − 14·53-s − 4·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.229·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.696·33-s − 0.676·35-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 1.75·47-s + 9/7·49-s + 0.280·51-s − 1.92·53-s − 0.539·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 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
−8.048969399455866367430327486185, −7.12849204514808959615817283020, −6.66235577427066409912732404574, −5.71052096351149678347394611825, −5.18263232888194878571069280029, −3.96714232191302426652690505064, −3.13456954226054422635167828519, −2.72575868345849785488559068724, −1.47535115843427244211568232535, 0,
1.47535115843427244211568232535, 2.72575868345849785488559068724, 3.13456954226054422635167828519, 3.96714232191302426652690505064, 5.18263232888194878571069280029, 5.71052096351149678347394611825, 6.66235577427066409912732404574, 7.12849204514808959615817283020, 8.048969399455866367430327486185
