| L(s) = 1 | + (−1.90 − 0.618i)2-s + (2.86 − 3.94i)3-s + (3.23 + 2.35i)4-s + (7.54 + 8.24i)5-s + (−7.89 + 5.73i)6-s − 32.3i·7-s + (−4.70 − 6.47i)8-s + (0.977 + 3.00i)9-s + (−9.25 − 20.3i)10-s + (14.5 − 44.7i)11-s + (18.5 − 6.03i)12-s + (9.48 − 3.08i)13-s + (−20.0 + 61.5i)14-s + (54.2 − 6.13i)15-s + (4.94 + 15.2i)16-s + (26.8 + 36.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 − 0.218i)2-s + (0.552 − 0.760i)3-s + (0.404 + 0.293i)4-s + (0.675 + 0.737i)5-s + (−0.537 + 0.390i)6-s − 1.74i·7-s + (−0.207 − 0.286i)8-s + (0.0361 + 0.111i)9-s + (−0.292 − 0.643i)10-s + (0.398 − 1.22i)11-s + (0.446 − 0.145i)12-s + (0.202 − 0.0657i)13-s + (−0.382 + 1.17i)14-s + (0.933 − 0.105i)15-s + (0.0772 + 0.237i)16-s + (0.383 + 0.527i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.11511 - 0.679156i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11511 - 0.679156i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-7.54 - 8.24i)T \) |
| good | 3 | \( 1 + (-2.86 + 3.94i)T + (-8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 + 32.3iT - 343T^{2} \) |
| 11 | \( 1 + (-14.5 + 44.7i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-9.48 + 3.08i)T + (1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-26.8 - 36.9i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (94.7 - 68.8i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-8.30 - 2.69i)T + (9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-188. - 137. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (211. - 154. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-14.1 + 4.58i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (79.2 + 244. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 81.5iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (128. - 177. i)T + (-3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (190. - 262. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-162. - 498. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-77.8 + 239. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (435. + 599. i)T + (-9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-306. - 222. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (349. + 113. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-741. - 538. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (258. + 355. i)T + (-1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-117. + 363. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-395. + 543. i)T + (-2.82e5 - 8.68e5i)T^{2} \) |
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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
−14.37122925195045756056672438594, −13.85300241476847561424162801614, −12.77152600874395048843761387064, −10.80208914344373484258500133914, −10.41122647054542633305840489320, −8.619475216019780682111226724387, −7.45139023993898022394721313134, −6.43650291127377391685241737369, −3.44054675844311510826192528901, −1.43849610474116296664429089239,
2.28217014084241073146945873088, 4.82299445654942632730976950943, 6.34119062090657584713671367448, 8.470143755744495087952507071144, 9.261840722995717049010199262362, 9.875392075729709981408275136340, 11.79141158794689746523049226038, 12.83899599794852635416360160262, 14.66355103772124426151815931032, 15.27748055677948752689955215411
