| L(s) = 1 | + (−2.10 + 3.65i)2-s + (−5.18 − 0.288i)3-s + (−4.88 − 8.45i)4-s − 7.82·5-s + (11.9 − 18.3i)6-s + (16.1 − 9.02i)7-s + 7.43·8-s + (26.8 + 2.99i)9-s + (16.4 − 28.5i)10-s − 12.8·11-s + (22.8 + 45.2i)12-s + (6.74 − 11.6i)13-s + (−1.15 + 78.0i)14-s + (40.5 + 2.25i)15-s + (23.3 − 40.5i)16-s + (35.5 − 61.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.745 + 1.29i)2-s + (−0.998 − 0.0555i)3-s + (−0.610 − 1.05i)4-s − 0.699·5-s + (0.815 − 1.24i)6-s + (0.873 − 0.487i)7-s + 0.328·8-s + (0.993 + 0.110i)9-s + (0.521 − 0.903i)10-s − 0.352·11-s + (0.550 + 1.08i)12-s + (0.143 − 0.249i)13-s + (−0.0220 + 1.48i)14-s + (0.698 + 0.0388i)15-s + (0.365 − 0.632i)16-s + (0.506 − 0.877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.372938 - 0.102902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.372938 - 0.102902i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.18 + 0.288i)T \) |
| 7 | \( 1 + (-16.1 + 9.02i)T \) |
| good | 2 | \( 1 + (2.10 - 3.65i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + 7.82T + 125T^{2} \) |
| 11 | \( 1 + 12.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-6.74 + 11.6i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-35.5 + 61.4i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (46.2 + 80.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 - 3.94T + 1.21e4T^{2} \) |
| 29 | \( 1 + (90.3 + 156. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (135. + 234. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-110. - 190. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (33.6 - 58.2i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-237. - 411. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-256. + 444. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (238. - 413. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (358. + 621. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (188. - 326. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (347. + 601. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 230.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (258. - 446. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-471. + 817. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (84.4 + 146. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (149. + 258. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-389. - 674. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.95495620297063440038420933592, −13.42440932148542580126568224742, −11.85187891351426430166914833984, −10.96844595901338246669734425706, −9.556223090997904306750173525653, −7.921103893346190095302725809402, −7.35315108299976086062645889412, −5.92863677445856507327534684429, −4.60600397299439443473270148354, −0.40093709247360137859326481668,
1.60956641895751210782408938548, 3.93408764158245774365307564208, 5.69499058902132694544580876256, 7.71979657255523763728814016310, 8.960724382902886021500216256769, 10.49220672805024046630818710939, 11.02055354082051730890612522639, 12.11026564578191215796666342598, 12.59007748765491005406064706954, 14.65753281782690013763559576464
