| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 4.47·13-s + 15-s + 4.47·17-s − 21-s − 2.47·23-s + 25-s − 27-s + 0.472·29-s + 2.47·31-s − 35-s + 0.472·37-s + 4.47·39-s − 6.94·41-s − 1.52·43-s − 45-s + 6.47·47-s + 49-s − 4.47·51-s − 2·53-s + 4·59-s − 3.52·61-s + 63-s + 4.47·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s − 1.24·13-s + 0.258·15-s + 1.08·17-s − 0.218·21-s − 0.515·23-s + 0.200·25-s − 0.192·27-s + 0.0876·29-s + 0.444·31-s − 0.169·35-s + 0.0776·37-s + 0.716·39-s − 1.08·41-s − 0.232·43-s − 0.149·45-s + 0.944·47-s + 0.142·49-s − 0.626·51-s − 0.274·53-s + 0.520·59-s − 0.451·61-s + 0.125·63-s + 0.554·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 - 6.47T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 0.944T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−7.58938889458375342114894915733, −7.04067241544700805494926835356, −6.19601871327697399192661268510, −5.40197774647185895400154717762, −4.85546647127553761811008252183, −4.11736709701848401100996505607, −3.22105032619658078629174650559, −2.25594340468474005481396056674, −1.16515863644149473107174368034, 0,
1.16515863644149473107174368034, 2.25594340468474005481396056674, 3.22105032619658078629174650559, 4.11736709701848401100996505607, 4.85546647127553761811008252183, 5.40197774647185895400154717762, 6.19601871327697399192661268510, 7.04067241544700805494926835356, 7.58938889458375342114894915733
