| L(s) = 1 | + (1.90 + 0.618i)2-s + (−5.11 + 7.04i)3-s + (3.23 + 2.35i)4-s + (−8.52 − 7.23i)5-s + (−14.0 + 10.2i)6-s + 24.3i·7-s + (4.70 + 6.47i)8-s + (−15.0 − 46.4i)9-s + (−11.7 − 19.0i)10-s + (−11.9 + 36.8i)11-s + (−33.1 + 10.7i)12-s + (66.0 − 21.4i)13-s + (−15.0 + 46.3i)14-s + (94.5 − 22.9i)15-s + (4.94 + 15.2i)16-s + (49.3 + 67.9i)17-s + ⋯ |
| L(s) = 1 | + (0.672 + 0.218i)2-s + (−0.984 + 1.35i)3-s + (0.404 + 0.293i)4-s + (−0.762 − 0.647i)5-s + (−0.958 + 0.696i)6-s + 1.31i·7-s + (0.207 + 0.286i)8-s + (−0.558 − 1.71i)9-s + (−0.371 − 0.601i)10-s + (−0.328 + 1.01i)11-s + (−0.796 + 0.258i)12-s + (1.40 − 0.457i)13-s + (−0.287 + 0.884i)14-s + (1.62 − 0.395i)15-s + (0.0772 + 0.237i)16-s + (0.704 + 0.969i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.498275 + 1.11124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.498275 + 1.11124i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.90 - 0.618i)T \) |
| 5 | \( 1 + (8.52 + 7.23i)T \) |
| good | 3 | \( 1 + (5.11 - 7.04i)T + (-8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 - 24.3iT - 343T^{2} \) |
| 11 | \( 1 + (11.9 - 36.8i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-66.0 + 21.4i)T + (1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-49.3 - 67.9i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-65.5 + 47.6i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (68.3 + 22.2i)T + (9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (122. + 88.9i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-44.1 + 32.0i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (13.5 - 4.38i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-62.9 - 193. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 164. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-91.2 + 125. i)T + (-3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-114. + 158. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (153. + 472. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-83.4 + 256. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-160. - 220. i)T + (-9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-323. - 234. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-1.09e3 - 354. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-145. - 105. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (623. + 858. i)T + (-1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-1.16 + 3.57i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (254. - 349. i)T + (-2.82e5 - 8.68e5i)T^{2} \) |
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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
−15.58736024769289207920246541646, −14.96199641308738602963643403868, −12.86083331252101515352916841023, −11.95209279421375188755661027444, −11.09652073066642857351009041716, −9.637044754329399773612319683398, −8.222829319351407367829834561034, −5.94450086098823877561332150084, −5.05825811799477935459098462750, −3.77366720897454309554570176502,
0.912037618857759896099357681518, 3.62457656625821367056938027408, 5.75454181582715752095667712659, 6.94868074187924727261793997255, 7.80015158805023984970670705226, 10.67182706615744645550561628050, 11.31315956338867304390271537297, 12.22264553889501896100042434787, 13.69753794931676532447071766749, 13.91567760073216797170981714170
