| L(s) = 1 | + 7·5-s + 8.66i·7-s − 8.66i·11-s − 20·13-s + 8·17-s + 10.3i·19-s − 3.46i·23-s + 24·25-s + 10·29-s + 53.6i·31-s + 60.6i·35-s + 10·37-s + 50·41-s + 17.3i·43-s + 86.6i·47-s + ⋯ |
| L(s) = 1 | + 1.40·5-s + 1.23i·7-s − 0.787i·11-s − 1.53·13-s + 0.470·17-s + 0.546i·19-s − 0.150i·23-s + 0.959·25-s + 0.344·29-s + 1.73i·31-s + 1.73i·35-s + 0.270·37-s + 1.21·41-s + 0.402i·43-s + 1.84i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.126923820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126923820\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 7T + 25T^{2} \) |
| 7 | \( 1 - 8.66iT - 49T^{2} \) |
| 11 | \( 1 + 8.66iT - 121T^{2} \) |
| 13 | \( 1 + 20T + 169T^{2} \) |
| 17 | \( 1 - 8T + 289T^{2} \) |
| 19 | \( 1 - 10.3iT - 361T^{2} \) |
| 23 | \( 1 + 3.46iT - 529T^{2} \) |
| 29 | \( 1 - 10T + 841T^{2} \) |
| 31 | \( 1 - 53.6iT - 961T^{2} \) |
| 37 | \( 1 - 10T + 1.36e3T^{2} \) |
| 41 | \( 1 - 50T + 1.68e3T^{2} \) |
| 43 | \( 1 - 17.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 86.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 47T + 2.80e3T^{2} \) |
| 59 | \( 1 - 34.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 64T + 3.72e3T^{2} \) |
| 67 | \( 1 - 86.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 55T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.92iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 29.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 10T + 7.92e3T^{2} \) |
| 97 | \( 1 + 25T + 9.40e3T^{2} \) |
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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
−9.388104277504633719101175112347, −8.723347677217980754972354640026, −7.84778689898979465516301866012, −6.77766517750145552057146715746, −5.86974123300975736041289163675, −5.52646400875996070403503642147, −4.62217651849477793290517598416, −2.98206749161661123479611721392, −2.45680611554738645174159904311, −1.33912834401758181787889830134,
0.53898566033875193644866332684, 1.87219792794748056304243836089, 2.63649119869623753249201443633, 4.03435761503833530934764785295, 4.86640217784393146651632136794, 5.63073043113444835025137551537, 6.66327909914997678843776846573, 7.26460278554775923214984762589, 7.965923306154184043372105841496, 9.340506655936745197019453413042
