| L(s) = 1 | + (−0.158 + 1.25i)2-s + (0.573 − 0.474i)3-s + (0.393 + 0.101i)4-s + (−0.459 − 2.18i)5-s + (0.503 + 0.793i)6-s + (4.01 − 1.30i)7-s + (−1.11 + 2.82i)8-s + (−0.458 + 2.40i)9-s + (2.81 − 0.229i)10-s + (−4.86 − 0.614i)11-s + (0.273 − 0.128i)12-s + (−0.601 − 0.114i)13-s + (0.997 + 5.23i)14-s + (−1.30 − 1.03i)15-s + (−2.64 − 1.45i)16-s + (0.0971 + 0.378i)17-s + ⋯ |
| L(s) = 1 | + (−0.111 + 0.885i)2-s + (0.331 − 0.274i)3-s + (0.196 + 0.0505i)4-s + (−0.205 − 0.978i)5-s + (0.205 + 0.324i)6-s + (1.51 − 0.492i)7-s + (−0.395 + 0.998i)8-s + (−0.152 + 0.800i)9-s + (0.889 − 0.0725i)10-s + (−1.46 − 0.185i)11-s + (0.0790 − 0.0371i)12-s + (−0.166 − 0.0318i)13-s + (0.266 + 1.39i)14-s + (−0.336 − 0.267i)15-s + (−0.662 − 0.364i)16-s + (0.0235 + 0.0917i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16506 + 0.398279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16506 + 0.398279i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.459 + 2.18i)T \) |
| good | 2 | \( 1 + (0.158 - 1.25i)T + (-1.93 - 0.497i)T^{2} \) |
| 3 | \( 1 + (-0.573 + 0.474i)T + (0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (-4.01 + 1.30i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (4.86 + 0.614i)T + (10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (0.601 + 0.114i)T + (12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (-0.0971 - 0.378i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (2.39 - 2.89i)T + (-3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (-4.06 + 4.33i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.409 + 6.50i)T + (-28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (4.18 - 1.07i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (-1.41 + 2.57i)T + (-19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (3.09 - 2.90i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (5.23 + 7.20i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-3.31 - 8.36i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (4.97 + 3.15i)T + (22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (-5.82 - 12.3i)T + (-37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (-0.950 - 0.892i)T + (3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-7.39 - 0.465i)T + (66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (-5.07 + 2.00i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (2.84 + 1.34i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (-7.76 - 9.38i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (-4.10 - 3.39i)T + (15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (-5.98 + 12.7i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (-9.23 + 0.581i)T + (96.2 - 12.1i)T^{2} \) |
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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
−13.66169059853985637432877064362, −12.66875298838532062783326693119, −11.37263427099889704288069487609, −10.56642179265234301438529811852, −8.541724129095950277718545251344, −8.069056614966089044467635209513, −7.41450198117426190113895904613, −5.52906771374309595110162168738, −4.72683938202607204208925666357, −2.15637615588792120748125178079,
2.21721810388591146314928776768, 3.35482470666059095857796625750, 5.14832113092080731131858915507, 6.82991278533442038895477717805, 8.002088851389189895069010737574, 9.302589879401337772486407742281, 10.51032736937296598150494395395, 11.14880373454299781722193549418, 11.89947802224315484804629336987, 13.09962987206227053960207520985
