| L(s) = 1 | + (−1.87 + 0.684i)2-s + (−1.18 − 0.429i)3-s + (3.06 − 2.57i)4-s + (0.00632 + 0.0358i)5-s + 2.51·6-s + (−0.240 − 1.36i)7-s + (−4.00 + 6.92i)8-s + (−19.4 − 16.3i)9-s + (−0.0363 − 0.0630i)10-s + (26.7 − 46.3i)11-s + (−4.72 + 1.71i)12-s + (58.2 − 48.8i)13-s + (1.38 + 2.39i)14-s + (0.00793 − 0.0450i)15-s + (2.77 − 15.7i)16-s + (−18.5 − 15.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (−0.227 − 0.0826i)3-s + (0.383 − 0.321i)4-s + (0.000565 + 0.00320i)5-s + 0.170·6-s + (−0.0129 − 0.0735i)7-s + (−0.176 + 0.306i)8-s + (−0.721 − 0.605i)9-s + (−0.00115 − 0.00199i)10-s + (0.733 − 1.27i)11-s + (−0.113 + 0.0413i)12-s + (1.24 − 1.04i)13-s + (0.0263 + 0.0457i)14-s + (0.000136 − 0.000775i)15-s + (0.0434 − 0.246i)16-s + (−0.264 − 0.221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.722600 - 0.493947i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.722600 - 0.493947i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 - 0.684i)T \) |
| 37 | \( 1 + (-203. - 96.5i)T \) |
| good | 3 | \( 1 + (1.18 + 0.429i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (-0.00632 - 0.0358i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (0.240 + 1.36i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-26.7 + 46.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-58.2 + 48.8i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (18.5 + 15.5i)T + (853. + 4.83e3i)T^{2} \) |
| 19 | \( 1 + (10.5 + 3.84i)T + (5.25e3 + 4.40e3i)T^{2} \) |
| 23 | \( 1 + (71.2 + 123. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (145. - 251. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 150.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (77.5 - 65.0i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 - 52.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + (100. + 173. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (25.9 - 147. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-3.38 + 19.2i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-156. + 131. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (117. + 667. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-487. - 177. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 - 7.48T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-95.0 - 539. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (1.08e3 + 913. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (195. - 1.10e3i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-729. - 1.26e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.05285887674131722946557952691, −12.69743240313096063001901097709, −11.33469155209534560729595210799, −10.67186475323965347294008154059, −8.992522275627560603523787605507, −8.362823906695325356120361894763, −6.61584575594030909064220661511, −5.70840082937435669381802215213, −3.35352974824783660524768093422, −0.74652109679099088612404387382,
1.89106472678425080753079461426, 4.08790361830951441049291841289, 6.00031241631555899230313568510, 7.37004791821477491240364748969, 8.745332997394990761017814985769, 9.688312569779647246095028258497, 11.08567147646424795943279465504, 11.71333866558149384729105597577, 13.08487561497054992110686313956, 14.30754796278811391083588000332
