| L(s) = 1 | + (0.939 + 0.342i)2-s + (−0.232 − 1.31i)3-s + (0.766 + 0.642i)4-s + (0.527 − 0.442i)5-s + (0.232 − 1.31i)6-s + (0.106 − 0.185i)7-s + (0.500 + 0.866i)8-s + (1.13 − 0.412i)9-s + (0.647 − 0.235i)10-s + (0.5 + 0.866i)11-s + (0.669 − 1.16i)12-s + (0.581 − 3.29i)13-s + (0.163 − 0.137i)14-s + (−0.706 − 0.593i)15-s + (0.173 + 0.984i)16-s + (−0.626 − 0.228i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (−0.134 − 0.761i)3-s + (0.383 + 0.321i)4-s + (0.235 − 0.198i)5-s + (0.0949 − 0.538i)6-s + (0.0404 − 0.0700i)7-s + (0.176 + 0.306i)8-s + (0.377 − 0.137i)9-s + (0.204 − 0.0745i)10-s + (0.150 + 0.261i)11-s + (0.193 − 0.334i)12-s + (0.161 − 0.914i)13-s + (0.0437 − 0.0367i)14-s + (−0.182 − 0.153i)15-s + (0.0434 + 0.246i)16-s + (−0.151 − 0.0553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.01327 - 0.588516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01327 - 0.588516i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-4.10 + 1.46i)T \) |
| good | 3 | \( 1 + (0.232 + 1.31i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-0.527 + 0.442i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.106 + 0.185i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-0.581 + 3.29i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.626 + 0.228i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (2.08 + 1.74i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.232 - 0.0845i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (0.155 - 0.269i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.31T + 37T^{2} \) |
| 41 | \( 1 + (-0.261 - 1.48i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (6.33 - 5.31i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (10.2 - 3.73i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-5.70 - 4.78i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-7.60 - 2.76i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (4.00 + 3.36i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (14.4 - 5.24i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.85 - 5.75i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (1.01 + 5.74i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.213 + 1.20i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (4.00 - 6.93i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.338 - 1.92i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-10.0 - 3.66i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.44843669886736817753519248569, −10.28074122261333336081795002267, −9.314653033109186621209830598407, −8.001651636148987673032852820590, −7.31573464453591934376955260374, −6.36791631360144073729713880514, −5.46279722433497649386633485959, −4.32846314125020266432492846181, −2.95161850471626581120843389805, −1.38022648628930259667142956940,
1.87783267499272930698121755075, 3.45897519931965917185421636224, 4.35436909324481001066358234636, 5.34147710348968064422702115379, 6.36565596130391251733097792828, 7.39547325093697574049088848012, 8.746976748316045329492475441914, 9.837487264642767279677502559914, 10.30467060231585595866447653821, 11.43716265271161579748860706375
