| L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (−0.959 + 0.281i)6-s + (4.15 + 2.67i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)10-s + (0.0610 + 0.424i)11-s + (−0.142 − 0.989i)12-s + (2.87 − 1.84i)13-s + (−3.23 + 3.73i)14-s + (0.415 − 0.909i)15-s + (0.841 + 0.540i)16-s + (−0.972 + 0.285i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.292 − 0.337i)5-s + (−0.391 + 0.115i)6-s + (1.57 + 1.01i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.266 − 0.170i)10-s + (0.0184 + 0.128i)11-s + (−0.0410 − 0.285i)12-s + (0.796 − 0.512i)13-s + (−0.865 + 0.998i)14-s + (0.107 − 0.234i)15-s + (0.210 + 0.135i)16-s + (−0.235 + 0.0692i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03208 + 1.30105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03208 + 1.30105i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (-4.78 + 0.239i)T \) |
| good | 7 | \( 1 + (-4.15 - 2.67i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.0610 - 0.424i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.87 + 1.84i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.972 - 0.285i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.62 - 0.476i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (5.78 - 1.69i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.844 - 1.84i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (5.19 - 5.99i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.09 - 4.73i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.20 - 4.83i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 + (10.9 + 7.06i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (6.23 - 4.00i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.99 + 8.75i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.06 + 7.38i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.922 + 6.41i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (0.920 + 0.270i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (1.94 - 1.24i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.00 + 5.78i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (3.74 + 8.19i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (2.89 + 3.33i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.86517778961047977033363204100, −9.519350993590853044436676532400, −8.790978367503405351364152603464, −8.225395246352313529339379325136, −7.51479353767322288018470343371, −6.09473931612803639904662553223, −5.15486242573412843428570559062, −4.64909227810077757787796498082, −3.27254573451074023643338696570, −1.59137874884699765580430112211,
1.04151436511468224957935935736, 2.11983020970895926108356195512, 3.60908451961668273141153504859, 4.39264044752530491058747648678, 5.58365759108162276076623590287, 7.07778699186400655846861293234, 7.59807991735570005976954079217, 8.518252523254397572201419229286, 9.263499459532707477621494621505, 10.62668444034402001338586424519
