| L(s) = 1 | + 2·3-s + 2·5-s + 7-s + 9-s + 2·11-s − 4·13-s + 4·15-s − 6·17-s − 4·19-s + 2·21-s − 23-s − 25-s − 4·27-s − 6·29-s − 4·31-s + 4·33-s + 2·35-s + 8·37-s − 8·39-s + 6·41-s − 6·43-s + 2·45-s − 8·47-s + 49-s − 12·51-s + 4·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 1.03·15-s − 1.45·17-s − 0.917·19-s + 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s + 0.338·35-s + 1.31·37-s − 1.28·39-s + 0.937·41-s − 0.914·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s − 1.68·51-s + 0.549·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.92489068434041, −16.51337609938092, −15.31223516512347, −15.11723611694062, −14.58048004526530, −14.02807684846854, −13.60986041829871, −12.92662322615946, −12.56628760631929, −11.45896578405788, −11.18570294227953, −10.28494408073950, −9.571884607620886, −9.239457052549565, −8.749120720062985, −7.944998771598409, −7.478549307776074, −6.581756466965419, −6.071100541737439, −5.183075198944190, −4.411093756540659, −3.795546442393571, −2.761173194431555, −2.136941713222546, −1.722510272521193, 0,
1.722510272521193, 2.136941713222546, 2.761173194431555, 3.795546442393571, 4.411093756540659, 5.183075198944190, 6.071100541737439, 6.581756466965419, 7.478549307776074, 7.944998771598409, 8.749120720062985, 9.239457052549565, 9.571884607620886, 10.28494408073950, 11.18570294227953, 11.45896578405788, 12.56628760631929, 12.92662322615946, 13.60986041829871, 14.02807684846854, 14.58048004526530, 15.11723611694062, 15.31223516512347, 16.51337609938092, 16.92489068434041
