| L(s) = 1 | − 3-s − 2·5-s + 7-s − 2·9-s − 2·11-s + 13-s + 2·15-s − 6·19-s − 21-s − 25-s + 5·27-s − 29-s + 31-s + 2·33-s − 2·35-s − 6·37-s − 39-s + 3·41-s + 4·45-s − 3·47-s + 49-s + 6·53-s + 4·55-s + 6·57-s − 8·59-s − 10·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s − 2/3·9-s − 0.603·11-s + 0.277·13-s + 0.516·15-s − 1.37·19-s − 0.218·21-s − 1/5·25-s + 0.962·27-s − 0.185·29-s + 0.179·31-s + 0.348·33-s − 0.338·35-s − 0.986·37-s − 0.160·39-s + 0.468·41-s + 0.596·45-s − 0.437·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s + 0.794·57-s − 1.04·59-s − 1.28·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.15754613571073, −12.42172513111590, −12.21567952557029, −11.74369404978783, −11.20368558910672, −10.96393501157926, −10.46730372299098, −10.12992319306523, −9.266477098448727, −8.842166130994534, −8.321865696034464, −8.082035176059164, −7.469779332889060, −7.022188795010408, −6.383065547067124, −5.911984225015796, −5.535345867323483, −4.809334035092667, −4.502219057982150, −3.919699702233896, −3.308997590064325, −2.766144051273932, −2.096323275346269, −1.446852469945156, −0.5082868968073548, 0,
0.5082868968073548, 1.446852469945156, 2.096323275346269, 2.766144051273932, 3.308997590064325, 3.919699702233896, 4.502219057982150, 4.809334035092667, 5.535345867323483, 5.911984225015796, 6.383065547067124, 7.022188795010408, 7.469779332889060, 8.082035176059164, 8.321865696034464, 8.842166130994534, 9.266477098448727, 10.12992319306523, 10.46730372299098, 10.96393501157926, 11.20368558910672, 11.74369404978783, 12.21567952557029, 12.42172513111590, 13.15754613571073
