| 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
−10.37575823197706783594530940086, −10.05924926902608298583328330475, −9.430354710539475654567969010913, −7.78544637157994310256402204764, −6.55480765083933884364561520550, −5.78580046017651311010541180671, −4.82385090427556172929452110795, −4.12256544364769165832972314819, −2.50813860605122894222567993533, −1.28258354909276188732014596157,
2.08245616338343156989866305542, 2.97040597210734596528728555772, 5.00545421883585463900273304689, 5.42135622090807729906793398885, 6.13135230943424359607543593238, 7.22288507419643404429799306120, 8.009352977787083838420434779712, 9.229285489360446025125699913190, 10.01512139905701869368742147329, 11.23168421843388302197353058992
