| L(s) = 1 | − 19.1·5-s − 40.5·11-s − 50.4·13-s − 51.9·17-s + 33.1·19-s − 62.8·23-s + 241.·25-s − 129.·29-s − 242.·31-s − 389.·37-s − 470.·41-s − 125.·43-s − 386.·47-s + 611.·53-s + 777.·55-s + 226.·59-s − 725.·61-s + 966.·65-s + 1.04e3·67-s − 169.·71-s + 381.·73-s − 1.16e3·79-s − 808.·83-s + 993.·85-s − 319.·89-s − 634.·95-s − 1.13e3·97-s + ⋯ |
| L(s) = 1 | − 1.71·5-s − 1.11·11-s − 1.07·13-s − 0.740·17-s + 0.400·19-s − 0.569·23-s + 1.93·25-s − 0.831·29-s − 1.40·31-s − 1.73·37-s − 1.79·41-s − 0.443·43-s − 1.19·47-s + 1.58·53-s + 1.90·55-s + 0.499·59-s − 1.52·61-s + 1.84·65-s + 1.90·67-s − 0.283·71-s + 0.612·73-s − 1.65·79-s − 1.06·83-s + 1.26·85-s − 0.380·89-s − 0.684·95-s − 1.18·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.03946074515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03946074515\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 19.1T + 125T^{2} \) |
| 11 | \( 1 + 40.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 129.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 389.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 386.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 611.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 226.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 725.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 169.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 381.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 808.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 319.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3T + 9.12e5T^{2} \) |
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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
−8.659693117894330352797763165182, −8.155997719563367634418770034158, −7.26692002035067058765301221032, −7.00691452005835200964035291488, −5.43974231843399539663464174009, −4.83951509392946619773653947635, −3.86591085740050503990022995277, −3.13160133917286455480368985379, −1.95010749013294888777710244899, −0.088066992600517137721565747356,
0.088066992600517137721565747356, 1.95010749013294888777710244899, 3.13160133917286455480368985379, 3.86591085740050503990022995277, 4.83951509392946619773653947635, 5.43974231843399539663464174009, 7.00691452005835200964035291488, 7.26692002035067058765301221032, 8.155997719563367634418770034158, 8.659693117894330352797763165182
