| L(s) = 1 | − 5-s + 13-s − 2·17-s + 4·19-s − 4·23-s + 25-s − 6·29-s + 8·31-s + 6·37-s + 2·41-s − 4·43-s − 7·49-s + 6·53-s + 2·61-s − 65-s − 8·67-s + 6·73-s − 4·79-s − 12·83-s + 2·85-s − 6·89-s − 4·95-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.277·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.312·41-s − 0.609·43-s − 49-s + 0.824·53-s + 0.256·61-s − 0.124·65-s − 0.977·67-s + 0.702·73-s − 0.450·79-s − 1.31·83-s + 0.216·85-s − 0.635·89-s − 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.17862538610672, −14.60947990337713, −14.11834596994194, −13.42555885260817, −13.22032723410531, −12.42814654384547, −11.95963962183872, −11.38202441591442, −11.13959140005506, −10.29672515469451, −9.840799006508953, −9.322749540787879, −8.649972897597634, −8.097136757351226, −7.672331964032545, −7.012094700516681, −6.432032161650790, −5.810389777043967, −5.232093359521670, −4.427537840176121, −4.038402367132823, −3.244766613318948, −2.656552183066255, −1.784508798103136, −0.9679331609666141, 0,
0.9679331609666141, 1.784508798103136, 2.656552183066255, 3.244766613318948, 4.038402367132823, 4.427537840176121, 5.232093359521670, 5.810389777043967, 6.432032161650790, 7.012094700516681, 7.672331964032545, 8.097136757351226, 8.649972897597634, 9.322749540787879, 9.840799006508953, 10.29672515469451, 11.13959140005506, 11.38202441591442, 11.95963962183872, 12.42814654384547, 13.22032723410531, 13.42555885260817, 14.11834596994194, 14.60947990337713, 15.17862538610672
