L(s) = 1 | + (−2.08 − 3.60i)5-s + (−18.2 − 2.97i)7-s + (33.1 + 19.1i)11-s + 49.6i·13-s + (7.65 − 13.2i)17-s + (122. − 70.9i)19-s + (−136. + 78.9i)23-s + (53.8 − 93.2i)25-s − 204. i·29-s + (−90.5 − 52.2i)31-s + (27.3 + 72.1i)35-s + (−194. − 336. i)37-s − 325.·41-s + 191.·43-s + (−249. − 432. i)47-s + ⋯ |
L(s) = 1 | + (−0.186 − 0.322i)5-s + (−0.986 − 0.160i)7-s + (0.909 + 0.524i)11-s + 1.05i·13-s + (0.109 − 0.189i)17-s + (1.48 − 0.856i)19-s + (−1.24 + 0.715i)23-s + (0.430 − 0.745i)25-s − 1.31i·29-s + (−0.524 − 0.302i)31-s + (0.131 + 0.348i)35-s + (−0.862 − 1.49i)37-s − 1.23·41-s + 0.678·43-s + (−0.775 − 1.34i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.168745984\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.168745984\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (18.2 + 2.97i)T \) |
good | 5 | \( 1 + (2.08 + 3.60i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-33.1 - 19.1i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 49.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-7.65 + 13.2i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-122. + 70.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (136. - 78.9i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 204. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (90.5 + 52.2i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (194. + 336. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 325.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 191.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (249. + 432. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (37.8 + 21.8i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-86.5 + 149. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-208. + 120. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-440. + 763. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.01e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (361. + 208. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-237. - 411. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 652.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (298. + 517. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.77e3iT - 9.12e5T^{2} \) |
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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
−9.903149150123110971338534252979, −9.558773677237026694986115296789, −8.640363786893782202595649029690, −7.33268560170485545562437637724, −6.72932881552836132618672708040, −5.62312933668780155179910807791, −4.34484484943803377433879270750, −3.51182071382386997445352939883, −1.97519176819536008599066683272, −0.39650433530372847593937931933,
1.22846745602065840342107689333, 3.11246350185650680169003992138, 3.60431154357891651982155072786, 5.26362541051279950105672729423, 6.16451646146071316826820983111, 7.02475812967646031082021380608, 8.067907686496435937817681833134, 9.010788694092836533213677608072, 9.947841567207065357938147883430, 10.59175397437286025622801239226