| L(s) = 1 | + (−0.330 − 0.943i)2-s + (−0.637 + 1.82i)3-s + (−0.781 + 0.623i)4-s + (1.50 − 1.65i)5-s + 1.93·6-s + (0.527 + 0.527i)7-s + (0.846 + 0.532i)8-s + (−0.569 − 0.453i)9-s + (−2.05 − 0.870i)10-s + (−0.835 + 1.04i)11-s + (−0.637 − 1.82i)12-s + (1.68 − 2.67i)13-s + (0.323 − 0.672i)14-s + (2.06 + 3.79i)15-s + (0.222 − 0.974i)16-s + (1.36 + 2.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.233 − 0.667i)2-s + (−0.368 + 1.05i)3-s + (−0.390 + 0.311i)4-s + (0.671 − 0.740i)5-s + 0.788·6-s + (0.199 + 0.199i)7-s + (0.299 + 0.188i)8-s + (−0.189 − 0.151i)9-s + (−0.651 − 0.275i)10-s + (−0.251 + 0.315i)11-s + (−0.184 − 0.526i)12-s + (0.466 − 0.743i)13-s + (0.0865 − 0.179i)14-s + (0.532 + 0.979i)15-s + (0.0556 − 0.243i)16-s + (0.329 + 0.525i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23889 + 0.118297i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23889 + 0.118297i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.330 + 0.943i)T \) |
| 5 | \( 1 + (-1.50 + 1.65i)T \) |
| 43 | \( 1 + (-6.46 + 1.09i)T \) |
| good | 3 | \( 1 + (0.637 - 1.82i)T + (-2.34 - 1.87i)T^{2} \) |
| 7 | \( 1 + (-0.527 - 0.527i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.835 - 1.04i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.68 + 2.67i)T + (-5.64 - 11.7i)T^{2} \) |
| 17 | \( 1 + (-1.36 - 2.16i)T + (-7.37 + 15.3i)T^{2} \) |
| 19 | \( 1 + (-2.35 - 2.95i)T + (-4.22 + 18.5i)T^{2} \) |
| 23 | \( 1 + (1.13 - 0.127i)T + (22.4 - 5.11i)T^{2} \) |
| 29 | \( 1 + (-8.80 - 4.23i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-9.22 - 4.44i)T + (19.3 + 24.2i)T^{2} \) |
| 37 | \( 1 + (0.528 + 0.528i)T + 37iT^{2} \) |
| 41 | \( 1 + (-4.08 - 1.96i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (6.07 + 0.684i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (7.77 - 4.88i)T + (22.9 - 47.7i)T^{2} \) |
| 59 | \( 1 + (7.32 + 1.67i)T + (53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (4.05 + 8.41i)T + (-38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (0.388 + 0.0437i)T + (65.3 + 14.9i)T^{2} \) |
| 71 | \( 1 + (1.58 - 1.26i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (5.04 + 3.16i)T + (31.6 + 65.7i)T^{2} \) |
| 79 | \( 1 + 1.53iT - 79T^{2} \) |
| 83 | \( 1 + (8.92 + 3.12i)T + (64.8 + 51.7i)T^{2} \) |
| 89 | \( 1 + (0.881 - 0.424i)T + (55.4 - 69.5i)T^{2} \) |
| 97 | \( 1 + (-0.409 - 3.63i)T + (-94.5 + 21.5i)T^{2} \) |
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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
−10.87663979693049159424991236682, −10.21939169401525984161751363034, −9.738396862091112804997566703840, −8.685171497229826504928365437415, −7.921709912049739596719755500579, −6.12680652352964768560892314845, −5.13907109145096650192589699912, −4.45035524912372545846593076505, −3.09477791640977528447438830826, −1.41007884284574221406196844234,
1.12000664379925931278466463153, 2.69031017927214905162202275980, 4.52961376329608200778648242016, 5.94784270007425051273828327439, 6.44261720956467784134429315112, 7.28947660574878733913968236323, 8.068642521090146240487049303684, 9.321928872752873340209559561888, 10.12165926008174854653116098638, 11.21067584779709051397991224170
