| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 11-s + 2·14-s + 16-s − 20-s + 22-s − 4·23-s + 25-s + 2·28-s + 2·29-s − 4·31-s + 32-s − 2·35-s + 2·37-s − 40-s − 6·43-s + 44-s − 4·46-s − 3·49-s + 50-s + 6·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.534·14-s + 1/4·16-s − 0.223·20-s + 0.213·22-s − 0.834·23-s + 1/5·25-s + 0.377·28-s + 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.338·35-s + 0.328·37-s − 0.158·40-s − 0.914·43-s + 0.150·44-s − 0.589·46-s − 3/7·49-s + 0.141·50-s + 0.824·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.176417497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.176417497\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.82579252853734, −12.13325856403634, −11.84273485723813, −11.53187056263804, −11.00025172044418, −10.55419473774557, −10.13265860646285, −9.493890213813858, −9.018935180516036, −8.399853211056408, −8.039957970216789, −7.597664269261459, −7.095865762551403, −6.543566613415774, −6.127885093267568, −5.485231988509189, −5.079714621633498, −4.561602177768372, −4.027301823704362, −3.716102143893180, −2.963468824251327, −2.507761643140568, −1.674022584181179, −1.406730209779831, −0.4042557466429566,
0.4042557466429566, 1.406730209779831, 1.674022584181179, 2.507761643140568, 2.963468824251327, 3.716102143893180, 4.027301823704362, 4.561602177768372, 5.079714621633498, 5.485231988509189, 6.127885093267568, 6.543566613415774, 7.095865762551403, 7.597664269261459, 8.039957970216789, 8.399853211056408, 9.018935180516036, 9.493890213813858, 10.13265860646285, 10.55419473774557, 11.00025172044418, 11.53187056263804, 11.84273485723813, 12.13325856403634, 12.82579252853734
