L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.362 + 0.995i)3-s + (0.766 − 0.642i)4-s + (2.12 − 0.683i)5-s + 1.05i·6-s + (−1.06 + 0.187i)7-s + (0.500 − 0.866i)8-s + (1.43 + 1.20i)9-s + (1.76 − 1.37i)10-s + (0.436 − 0.755i)11-s + (0.362 + 0.995i)12-s + (1.18 − 0.992i)13-s + (−0.933 + 0.539i)14-s + (−0.0906 + 2.36i)15-s + (0.173 − 0.984i)16-s + (−0.626 − 0.525i)17-s + ⋯ |
L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.209 + 0.574i)3-s + (0.383 − 0.321i)4-s + (0.952 − 0.305i)5-s + 0.432i·6-s + (−0.401 + 0.0707i)7-s + (0.176 − 0.306i)8-s + (0.479 + 0.402i)9-s + (0.558 − 0.433i)10-s + (0.131 − 0.227i)11-s + (0.104 + 0.287i)12-s + (0.328 − 0.275i)13-s + (−0.249 + 0.144i)14-s + (−0.0234 + 0.611i)15-s + (0.0434 − 0.246i)16-s + (−0.151 − 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.12468 + 0.00861188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12468 + 0.00861188i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 5 | \( 1 + (-2.12 + 0.683i)T \) |
| 37 | \( 1 + (4.86 + 3.64i)T \) |
good | 3 | \( 1 + (0.362 - 0.995i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (1.06 - 0.187i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.436 + 0.755i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.18 + 0.992i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.626 + 0.525i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (0.570 - 1.56i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-2.32 - 4.02i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.84 + 1.06i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.18iT - 31T^{2} \) |
| 41 | \( 1 + (-1.27 + 1.06i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + 2.12T + 43T^{2} \) |
| 47 | \( 1 + (4.23 - 2.44i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.91 - 0.865i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (10.8 + 1.90i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.22 - 1.45i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (5.75 - 1.01i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (10.2 + 3.71i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + 10.5iT - 73T^{2} \) |
| 79 | \( 1 + (4.86 - 0.858i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.61 + 3.11i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (15.0 + 2.66i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-5.51 - 9.55i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.29976959097160963001327500348, −10.49541113450325090884686228945, −9.768881452152524971025998677044, −8.949282466589652391330715190052, −7.47861830419772272369952693156, −6.22933570685471172164154114155, −5.46412948375066257486489897270, −4.50987319611229669507191326991, −3.27212791262015223328164673938, −1.73721812485559845733912831369,
1.68727980633867582026239196669, 3.12288765424161898488161071604, 4.52097489333390051872455789509, 5.77499185710800774728414735136, 6.66484214711137428311084900790, 7.06831808905052333947076254965, 8.591309027894922565721656196857, 9.639193398069745548093031855046, 10.54124513519936248930016654757, 11.57544215346053582079091316512