| 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
−13.31887761555741826993596369558, −12.98236574687955616340883440513, −10.66862446999671502762529528285, −9.819817223827960067168917857747, −9.343514913612498729222460848742, −7.82732099958046823223091986182, −6.90124121478584985403622597801, −5.93038129634725635235037993717, −3.91710814162986135438786675395, −2.15329251566716511001043378549,
2.21212622813785794060467662527, 3.20878092133097400175198512903, 5.23686928848773348830532905687, 6.59172402488511011597025122915, 8.459257283359410311942633816423, 8.960242113059497643169652097624, 9.919332987560440930978378425481, 10.64219174745809116968476005545, 12.50782418388586721881579045542, 13.07687063605190143779678809947
