| L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−1.72 + 2.98i)5-s + (15.3 − 10.2i)7-s + 7.99·8-s + (−3.44 − 5.96i)10-s + (18.0 + 31.2i)11-s + 10.2·13-s + (2.44 + 36.9i)14-s + (−8 + 13.8i)16-s + (59.2 + 102. i)17-s + (19.3 − 33.4i)19-s + 13.7·20-s − 72.2·22-s + (−18.1 + 31.3i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.154 + 0.266i)5-s + (0.831 − 0.556i)7-s + 0.353·8-s + (−0.108 − 0.188i)10-s + (0.495 + 0.857i)11-s + 0.218·13-s + (0.0466 + 0.705i)14-s + (−0.125 + 0.216i)16-s + (0.845 + 1.46i)17-s + (0.233 − 0.404i)19-s + 0.154·20-s − 0.700·22-s + (−0.164 + 0.284i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.18548 + 0.790884i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18548 + 0.790884i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-15.3 + 10.2i)T \) |
| good | 5 | \( 1 + (1.72 - 2.98i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-18.0 - 31.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 10.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-59.2 - 102. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-19.3 + 33.4i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (18.1 - 31.3i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 12.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-72.7 - 126. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-0.685 + 1.18i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 168T + 6.89e4T^{2} \) |
| 43 | \( 1 - 299.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-251. + 435. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (312. + 541. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (21.1 + 36.5i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-219. + 380. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-381. - 661. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (289. + 501. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (471. - 816. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 474.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-410. + 711. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.10e3T + 9.12e5T^{2} \) |
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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
−13.25244762266693725373517925897, −11.99986780838520044144035756623, −10.83975581873744664745567028026, −9.969311005601626023268398801328, −8.651864754241102813060213678523, −7.64929337415567143061603601223, −6.71975851433823225849749421875, −5.25083037808514774902100032873, −3.89980772426093100416428577611, −1.45850402007738676730336206853,
1.01532438566663772465764155964, 2.83641656124131941327720682150, 4.47221414244489050935324096215, 5.84087164261801601127020526136, 7.61696386326996933913805589159, 8.575193991709920056796504425158, 9.475832772764178533294013743398, 10.78971021700093047021481947104, 11.72582445522837980372797193819, 12.30964374216570151191256759763
