| L(s) = 1 | + (1.10 + 0.879i)2-s + (−0.755 − 0.654i)3-s + (0.454 + 1.94i)4-s + (3.07 + 0.442i)5-s + (−0.261 − 1.38i)6-s + (1.27 + 2.78i)7-s + (−1.20 + 2.55i)8-s + (0.142 + 0.989i)9-s + (3.01 + 3.19i)10-s + (−0.100 − 0.340i)11-s + (0.932 − 1.76i)12-s + (−5.09 − 2.32i)13-s + (−1.04 + 4.20i)14-s + (−2.03 − 2.34i)15-s + (−3.58 + 1.76i)16-s + (4.48 − 2.88i)17-s + ⋯ |
| L(s) = 1 | + (0.783 + 0.621i)2-s + (−0.436 − 0.378i)3-s + (0.227 + 0.973i)4-s + (1.37 + 0.197i)5-s + (−0.106 − 0.567i)6-s + (0.481 + 1.05i)7-s + (−0.427 + 0.904i)8-s + (0.0474 + 0.329i)9-s + (0.954 + 1.00i)10-s + (−0.0301 − 0.102i)11-s + (0.269 − 0.510i)12-s + (−1.41 − 0.645i)13-s + (−0.278 + 1.12i)14-s + (−0.525 − 0.606i)15-s + (−0.896 + 0.442i)16-s + (1.08 − 0.699i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.85267 + 1.49778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85267 + 1.49778i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.10 - 0.879i)T \) |
| 3 | \( 1 + (0.755 + 0.654i)T \) |
| 23 | \( 1 + (3.41 - 3.36i)T \) |
| good | 5 | \( 1 + (-3.07 - 0.442i)T + (4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.27 - 2.78i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (0.100 + 0.340i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (5.09 + 2.32i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.48 + 2.88i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.803 + 1.25i)T + (-7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.99 - 6.21i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (1.04 + 1.20i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-5.87 + 0.844i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.40 + 9.74i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.06 - 4.38i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + (-6.17 + 2.82i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (12.5 + 5.73i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-5.74 + 4.98i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (0.617 - 2.10i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-8.31 - 2.44i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (6.36 + 4.09i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.12 + 11.2i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.513 - 0.0738i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (5.20 - 6.00i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.491 + 3.41i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.23168421843388302197353058992, −10.01512139905701869368742147329, −9.229285489360446025125699913190, −8.009352977787083838420434779712, −7.22288507419643404429799306120, −6.13135230943424359607543593238, −5.42135622090807729906793398885, −5.00545421883585463900273304689, −2.97040597210734596528728555772, −2.08245616338343156989866305542,
1.28258354909276188732014596157, 2.50813860605122894222567993533, 4.12256544364769165832972314819, 4.82385090427556172929452110795, 5.78580046017651311010541180671, 6.55480765083933884364561520550, 7.78544637157994310256402204764, 9.430354710539475654567969010913, 10.05924926902608298583328330475, 10.37575823197706783594530940086
