L(s) = 1 | + (1.64 − 0.553i)3-s + (−0.263 + 0.455i)5-s + (−0.333 − 2.62i)7-s + (2.38 − 1.81i)9-s + (−2.30 − 3.99i)11-s + (0.244 + 0.423i)13-s + (−0.179 + 0.893i)15-s + (2.75 − 4.77i)17-s + (1.83 + 3.18i)19-s + (−1.99 − 4.12i)21-s + (0.0269 − 0.0467i)23-s + (2.36 + 4.09i)25-s + (2.91 − 4.30i)27-s + (−3.28 + 5.68i)29-s + 6.07·31-s + ⋯ |
L(s) = 1 | + (0.947 − 0.319i)3-s + (−0.117 + 0.203i)5-s + (−0.125 − 0.992i)7-s + (0.796 − 0.605i)9-s + (−0.695 − 1.20i)11-s + (0.0678 + 0.117i)13-s + (−0.0464 + 0.230i)15-s + (0.668 − 1.15i)17-s + (0.421 + 0.730i)19-s + (−0.436 − 0.899i)21-s + (0.00562 − 0.00974i)23-s + (0.472 + 0.818i)25-s + (0.561 − 0.827i)27-s + (−0.609 + 1.05i)29-s + 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61489 - 0.916547i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61489 - 0.916547i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.64 + 0.553i)T \) |
| 7 | \( 1 + (0.333 + 2.62i)T \) |
good | 5 | \( 1 + (0.263 - 0.455i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.30 + 3.99i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.244 - 0.423i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.75 + 4.77i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.83 - 3.18i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0269 + 0.0467i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.28 - 5.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.07T + 31T^{2} \) |
| 37 | \( 1 + (-0.223 - 0.387i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.52 - 4.36i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.84 + 4.93i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.19T + 47T^{2} \) |
| 53 | \( 1 + (4.37 - 7.57i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 + 0.465T + 61T^{2} \) |
| 67 | \( 1 - 5.19T + 67T^{2} \) |
| 71 | \( 1 - 1.76T + 71T^{2} \) |
| 73 | \( 1 + (5.23 - 9.07i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 + (-4.49 + 7.78i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.05 + 12.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.22 + 9.04i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.68377164577002548005240550357, −9.843741780181742682700760718466, −8.957051380181308133105169389522, −7.88310473887077850441947283863, −7.42401393520945085037156292844, −6.35368346424915258121199813881, −5.00433685450599360009479533894, −3.57903686788314929630532667700, −2.96256578613130513993497855082, −1.10497819949197137355899448607,
2.01063493040734743396112598393, 3.00018800215054279687183016042, 4.33772690818246011275120367965, 5.23900951373941416975179039862, 6.52463701700576135250120352129, 7.82850859399769122050460954672, 8.271345227630379970710878007697, 9.403959176327348677324743149867, 9.898877754166843009165887999144, 10.89932616969772813081863731249