L(s) = 1 | + (8.21 − 2.66i)3-s + (−9.50 − 5.89i)5-s − 24.1i·7-s + (38.5 − 27.9i)9-s + (−18.1 − 13.1i)11-s + (29.0 + 40.0i)13-s + (−93.7 − 23.0i)15-s + (66.2 + 21.5i)17-s + (−40.4 + 124. i)19-s + (−64.5 − 198. i)21-s + (114. − 157. i)23-s + (55.5 + 111. i)25-s + (104. − 143. i)27-s + (20.0 + 61.6i)29-s + (47.6 − 146. i)31-s + ⋯ |
L(s) = 1 | + (1.58 − 0.513i)3-s + (−0.849 − 0.526i)5-s − 1.30i·7-s + (1.42 − 1.03i)9-s + (−0.496 − 0.361i)11-s + (0.620 + 0.854i)13-s + (−1.61 − 0.396i)15-s + (0.944 + 0.306i)17-s + (−0.487 + 1.50i)19-s + (−0.670 − 2.06i)21-s + (1.03 − 1.42i)23-s + (0.444 + 0.895i)25-s + (0.745 − 1.02i)27-s + (0.128 + 0.394i)29-s + (0.276 − 0.849i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.80999 - 1.28621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80999 - 1.28621i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (9.50 + 5.89i)T \) |
good | 3 | \( 1 + (-8.21 + 2.66i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 + 24.1iT - 343T^{2} \) |
| 11 | \( 1 + (18.1 + 13.1i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-29.0 - 40.0i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-66.2 - 21.5i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (40.4 - 124. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-114. + 157. i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-20.0 - 61.6i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-47.6 + 146. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-43.7 - 60.2i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (180. - 131. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 401. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (257. - 83.7i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (336. - 109. i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-557. + 404. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (168. + 122. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-587. - 190. i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (79.6 + 245. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-130. + 179. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-246. - 757. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (961. + 312. i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-398. - 289. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (1.32e3 - 431. i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−13.24437314788010902842723792767, −12.53899112977051981642501386962, −11.01201141164565067339728410227, −9.731894036202814851609925485652, −8.316591813256023486083412592148, −7.994691393124456382846061212553, −6.75214898694507279151870626429, −4.29418916604880578609704202680, −3.32750521715915191699294119687, −1.25257919766425361913876719269,
2.62220899814560927453009634763, 3.45018167316930305515041608324, 5.16460329400046113758675439274, 7.22884209294174133898708294862, 8.283905839591279766621652039999, 9.005362085623754883968892659467, 10.18463384333976872798566391681, 11.43446479739271934066817760841, 12.71293447665351544099755666583, 13.74807392083746655118205758893