| L(s) = 1 | − 32.3·5-s + 39.7·7-s − 105. i·11-s + 26.9i·13-s − 100. i·17-s + 382.·19-s + 717. i·23-s + 418.·25-s + 448. i·29-s + (−658. − 699. i)31-s − 1.28e3·35-s + 2.19e3i·37-s − 15.3·41-s − 1.30e3i·43-s + 250.·47-s + ⋯ |
| L(s) = 1 | − 1.29·5-s + 0.810·7-s − 0.873i·11-s + 0.159i·13-s − 0.346i·17-s + 1.05·19-s + 1.35i·23-s + 0.670·25-s + 0.532i·29-s + (−0.685 − 0.728i)31-s − 1.04·35-s + 1.60i·37-s − 0.00914·41-s − 0.706i·43-s + 0.113·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.519130623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.519130623\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 31 | \( 1 + (658. + 699. i)T \) |
| good | 5 | \( 1 + 32.3T + 625T^{2} \) |
| 7 | \( 1 - 39.7T + 2.40e3T^{2} \) |
| 11 | \( 1 + 105. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 26.9iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 100. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 382.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 717. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 448. iT - 7.07e5T^{2} \) |
| 37 | \( 1 - 2.19e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 15.3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.30e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 250.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 3.76e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 460.T + 1.21e7T^{2} \) |
| 61 | \( 1 + 2.52e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.82e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 1.57e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 2.84e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 8.01e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 5.38e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 7.70e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.70e3T + 8.85e7T^{2} \) |
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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
−9.038034561473169217882327270982, −8.172363279451474374367461968149, −7.68874787965167242407036870004, −6.89101876843937542358231585120, −5.60474307289036011365045513852, −4.86843686792565830217186498189, −3.79119760376308858124385807804, −3.13814187326454868320484113431, −1.57423323582549651353820104511, −0.46776626589176651117751856163,
0.74068023392993806014565041994, 2.00034480590913918539545656985, 3.26499697621448273555417461851, 4.28328373889892597928263395837, 4.83294476232770880980716147012, 5.99739596746482750813510472545, 7.28244597706745826651119611911, 7.60761147737829300791247546910, 8.450840779924445531705303047378, 9.257074712377473391014929562195
