| L(s) = 1 | + (−1.40 − 0.166i)2-s + (1.85 + 0.369i)3-s + (1.94 + 0.468i)4-s + (−2.19 + 0.402i)5-s + (−2.54 − 0.828i)6-s + (0.221 + 0.535i)7-s + (−2.65 − 0.981i)8-s + (0.540 + 0.223i)9-s + (3.15 − 0.199i)10-s + (6.32 − 1.25i)11-s + (3.43 + 1.58i)12-s + (3.59 + 5.38i)13-s + (−0.222 − 0.788i)14-s + (−4.23 − 0.0642i)15-s + (3.56 + 1.82i)16-s + (−0.825 + 0.825i)17-s + ⋯ |
| L(s) = 1 | + (−0.993 − 0.117i)2-s + (1.07 + 0.213i)3-s + (0.972 + 0.234i)4-s + (−0.983 + 0.180i)5-s + (−1.03 − 0.338i)6-s + (0.0837 + 0.202i)7-s + (−0.937 − 0.346i)8-s + (0.180 + 0.0746i)9-s + (0.998 − 0.0630i)10-s + (1.90 − 0.379i)11-s + (0.992 + 0.458i)12-s + (0.998 + 1.49i)13-s + (−0.0593 − 0.210i)14-s + (−1.09 − 0.0165i)15-s + (0.890 + 0.455i)16-s + (−0.200 + 0.200i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08281 + 0.306983i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08281 + 0.306983i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.166i)T \) |
| 5 | \( 1 + (2.19 - 0.402i)T \) |
| good | 3 | \( 1 + (-1.85 - 0.369i)T + (2.77 + 1.14i)T^{2} \) |
| 7 | \( 1 + (-0.221 - 0.535i)T + (-4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-6.32 + 1.25i)T + (10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-3.59 - 5.38i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (0.825 - 0.825i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.683 - 1.02i)T + (-7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-1.17 - 0.486i)T + (16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (1.56 + 0.311i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 7.46iT - 31T^{2} \) |
| 37 | \( 1 + (2.88 + 1.92i)T + (14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (1.15 - 2.78i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.849 + 0.169i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (-4.59 + 4.59i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.70 + 8.54i)T + (-48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (5.11 + 3.42i)T + (22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (-1.14 + 5.76i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (9.91 + 1.97i)T + (61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (12.6 - 5.23i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.96 + 7.16i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-4.66 + 4.66i)T - 79iT^{2} \) |
| 83 | \( 1 + (2.13 - 1.42i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-5.68 - 13.7i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 0.0145T + 97T^{2} \) |
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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
−11.63758053158511551223152438267, −10.83575429063136922772516911665, −9.395730571849604060893215993399, −8.840229925534107492318179335083, −8.377578396310590249408417782051, −7.09078176987777865068794775138, −6.35075093283996723801945563827, −3.97772403628750482453173881234, −3.36764437045721513858856780840, −1.64201901945466646942665164607,
1.18456365132426954086204795421, 2.96281474177124465971484463534, 4.00614705815268170056302630155, 5.97327460830094581669940963794, 7.24337887021214878814209753282, 7.84833505885668235016325903558, 8.809274487968161740252958912879, 9.194424991089402899697512861180, 10.58370459486803486365286684563, 11.43092513744661212510753353964
