| L(s) = 1 | + (−0.540 − 0.841i)2-s + (1.70 + 0.282i)3-s + (−0.415 + 0.909i)4-s + (2.84 − 0.836i)5-s + (−0.686 − 1.59i)6-s + (−3.02 + 2.61i)7-s + (0.989 − 0.142i)8-s + (2.84 + 0.964i)9-s + (−2.24 − 1.94i)10-s + (−2.79 − 1.79i)11-s + (−0.966 + 1.43i)12-s + (1.31 − 1.51i)13-s + (3.83 + 1.12i)14-s + (5.10 − 0.625i)15-s + (−0.654 − 0.755i)16-s + (−0.139 − 0.306i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 − 0.594i)2-s + (0.986 + 0.162i)3-s + (−0.207 + 0.454i)4-s + (1.27 − 0.374i)5-s + (−0.280 − 0.649i)6-s + (−1.14 + 0.990i)7-s + (0.349 − 0.0503i)8-s + (0.946 + 0.321i)9-s + (−0.709 − 0.614i)10-s + (−0.842 − 0.541i)11-s + (−0.279 + 0.414i)12-s + (0.365 − 0.421i)13-s + (1.02 + 0.301i)14-s + (1.31 − 0.161i)15-s + (−0.163 − 0.188i)16-s + (−0.0339 − 0.0742i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22945 - 0.290466i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22945 - 0.290466i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 + 0.841i)T \) |
| 3 | \( 1 + (-1.70 - 0.282i)T \) |
| 23 | \( 1 + (-0.384 + 4.78i)T \) |
| good | 5 | \( 1 + (-2.84 + 0.836i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (3.02 - 2.61i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (2.79 + 1.79i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.31 + 1.51i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.139 + 0.306i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (4.42 + 2.02i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (6.14 - 2.80i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.10 - 7.71i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (2.31 - 7.87i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (0.321 + 1.09i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.439 + 0.0632i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 8.33iT - 47T^{2} \) |
| 53 | \( 1 + (-0.581 - 0.671i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.38 - 2.93i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-7.37 + 1.05i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (6.97 + 10.8i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-6.79 - 10.5i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.56 + 9.98i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-11.8 - 10.2i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (5.91 + 1.73i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.63 + 11.3i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-1.88 - 6.41i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−13.07687063605190143779678809947, −12.50782418388586721881579045542, −10.64219174745809116968476005545, −9.919332987560440930978378425481, −8.960242113059497643169652097624, −8.459257283359410311942633816423, −6.59172402488511011597025122915, −5.23686928848773348830532905687, −3.20878092133097400175198512903, −2.21212622813785794060467662527,
2.15329251566716511001043378549, 3.91710814162986135438786675395, 5.93038129634725635235037993717, 6.90124121478584985403622597801, 7.82732099958046823223091986182, 9.343514913612498729222460848742, 9.819817223827960067168917857747, 10.66862446999671502762529528285, 12.98236574687955616340883440513, 13.31887761555741826993596369558
