| L(s) = 1 | − 2·3-s + 4·5-s − 7-s + 9-s − 13-s − 8·15-s − 17-s − 19-s + 2·21-s + 2·23-s + 11·25-s + 4·27-s + 8·31-s − 4·35-s − 2·37-s + 2·39-s − 43-s + 4·45-s + 13·47-s + 49-s + 2·51-s + 10·53-s + 2·57-s + 8·59-s − 8·61-s − 63-s − 4·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s − 2.06·15-s − 0.242·17-s − 0.229·19-s + 0.436·21-s + 0.417·23-s + 11/5·25-s + 0.769·27-s + 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.320·39-s − 0.152·43-s + 0.596·45-s + 1.89·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s + 0.264·57-s + 1.04·59-s − 1.02·61-s − 0.125·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.521176782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.521176782\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.09819467105870, −12.32584247537095, −12.02898946102250, −11.68327412321582, −10.79181333546245, −10.61219536938579, −10.29519882972329, −9.734468740419153, −9.315639048665546, −8.776609712085615, −8.457889692003666, −7.503743731117166, −7.007631815262219, −6.553148857565670, −6.140213803182349, −5.751088000119692, −5.399135620543528, −4.783121297934483, −4.443378873485153, −3.550330583479271, −2.722402289068471, −2.489585668246931, −1.752768576170039, −1.032567260336153, −0.5353538352983858,
0.5353538352983858, 1.032567260336153, 1.752768576170039, 2.489585668246931, 2.722402289068471, 3.550330583479271, 4.443378873485153, 4.783121297934483, 5.399135620543528, 5.751088000119692, 6.140213803182349, 6.553148857565670, 7.007631815262219, 7.503743731117166, 8.457889692003666, 8.776609712085615, 9.315639048665546, 9.734468740419153, 10.29519882972329, 10.61219536938579, 10.79181333546245, 11.68327412321582, 12.02898946102250, 12.32584247537095, 13.09819467105870
