| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 4·11-s + 2·13-s + 16-s + 4·17-s + 20-s − 4·22-s + 6·23-s + 25-s + 2·26-s − 10·29-s − 2·31-s + 32-s + 4·34-s + 2·37-s + 40-s − 6·41-s − 8·43-s − 4·44-s + 6·46-s − 6·47-s − 7·49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.970·17-s + 0.223·20-s − 0.852·22-s + 1.25·23-s + 1/5·25-s + 0.392·26-s − 1.85·29-s − 0.359·31-s + 0.176·32-s + 0.685·34-s + 0.328·37-s + 0.158·40-s − 0.937·41-s − 1.21·43-s − 0.603·44-s + 0.884·46-s − 0.875·47-s − 49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.01778549232522, −14.92156001326932, −14.27657494037546, −13.49215639347787, −13.20406012821167, −12.94589576140105, −12.23401938021150, −11.59909551303596, −11.03056792610756, −10.68026410123575, −9.947046024987295, −9.579131058005864, −8.763772750624295, −8.175127110336476, −7.591877250889322, −7.047899136101473, −6.397950157046024, −5.710805909138762, −5.231895454704758, −4.911826960164638, −3.877543879348923, −3.315594520387230, −2.778674662487902, −1.908450771304890, −1.263888695085272, 0,
1.263888695085272, 1.908450771304890, 2.778674662487902, 3.315594520387230, 3.877543879348923, 4.911826960164638, 5.231895454704758, 5.710805909138762, 6.397950157046024, 7.047899136101473, 7.591877250889322, 8.175127110336476, 8.763772750624295, 9.579131058005864, 9.947046024987295, 10.68026410123575, 11.03056792610756, 11.59909551303596, 12.23401938021150, 12.94589576140105, 13.20406012821167, 13.49215639347787, 14.27657494037546, 14.92156001326932, 15.01778549232522
