| L(s) = 1 | + (−0.988 − 0.149i)2-s + (0.826 − 0.563i)3-s + (0.955 + 0.294i)4-s + (−0.189 − 2.52i)5-s + (−0.900 + 0.433i)6-s + (2.49 + 0.893i)7-s + (−0.900 − 0.433i)8-s + (0.365 − 0.930i)9-s + (−0.189 + 2.52i)10-s + (−0.833 − 2.12i)11-s + (0.955 − 0.294i)12-s + (−0.709 + 0.890i)13-s + (−2.32 − 1.25i)14-s + (−1.57 − 1.97i)15-s + (0.826 + 0.563i)16-s + (1.51 + 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 − 0.105i)2-s + (0.477 − 0.325i)3-s + (0.477 + 0.147i)4-s + (−0.0846 − 1.12i)5-s + (−0.367 + 0.177i)6-s + (0.941 + 0.337i)7-s + (−0.318 − 0.153i)8-s + (0.121 − 0.310i)9-s + (−0.0598 + 0.798i)10-s + (−0.251 − 0.640i)11-s + (0.275 − 0.0850i)12-s + (−0.196 + 0.246i)13-s + (−0.622 − 0.335i)14-s + (−0.407 − 0.511i)15-s + (0.206 + 0.140i)16-s + (0.366 + 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.922183 - 0.676647i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.922183 - 0.676647i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.988 + 0.149i)T \) |
| 3 | \( 1 + (-0.826 + 0.563i)T \) |
| 7 | \( 1 + (-2.49 - 0.893i)T \) |
| good | 5 | \( 1 + (0.189 + 2.52i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (0.833 + 2.12i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (0.709 - 0.890i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.51 - 1.40i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (2.80 + 4.85i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.58 + 2.39i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.266 + 1.16i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (3.10 - 5.36i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.74 + 2.07i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-1.04 - 0.503i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-3.80 + 1.83i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (6.54 + 0.985i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (4.64 + 1.43i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (1.02 - 13.6i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-3.45 + 1.06i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (6.29 - 10.9i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.34 - 14.6i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (6.13 - 0.925i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-7.36 - 12.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.61 - 5.79i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (4.34 - 11.0i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 1.58T + 97T^{2} \) |
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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
−11.52675005676582390117202801608, −10.70918255981853135288649583969, −9.336465135411555144937629977784, −8.629420644003283451780992916227, −8.151465756755751282540443859115, −6.98755427019664631486588857029, −5.54081504975472161333404600980, −4.39239607327014323609947012358, −2.57944061231709487418916332172, −1.12772404086043256089980554673,
1.99520181198931519242428746980, 3.34379198997139383116397403411, 4.80035337034026370992895093728, 6.28005632966061014384167377222, 7.60240605823981387760166982755, 7.81805880729530491540131229728, 9.228867402437122889912925310374, 10.15161022843644280694588872584, 10.80039064514373694919891796583, 11.57510934454980678717527937958
