| L(s) = 1 | + (2.95 + 3.41i)2-s + (−2.52 − 1.62i)3-s + (−1.76 + 12.2i)4-s + (−6.65 + 14.5i)5-s + (−1.92 − 13.4i)6-s + (1.88 + 0.553i)7-s + (−16.6 + 10.6i)8-s + (3.73 + 8.18i)9-s + (−69.4 + 20.3i)10-s + (−16.3 + 18.9i)11-s + (24.3 − 28.0i)12-s + (54.6 − 16.0i)13-s + (3.68 + 8.06i)14-s + (40.4 − 25.9i)15-s + (9.45 + 2.77i)16-s + (7.57 + 52.6i)17-s + ⋯ |
| L(s) = 1 | + (1.04 + 1.20i)2-s + (−0.485 − 0.312i)3-s + (−0.220 + 1.53i)4-s + (−0.595 + 1.30i)5-s + (−0.131 − 0.912i)6-s + (0.101 + 0.0298i)7-s + (−0.734 + 0.471i)8-s + (0.138 + 0.303i)9-s + (−2.19 + 0.644i)10-s + (−0.449 + 0.518i)11-s + (0.584 − 0.674i)12-s + (1.16 − 0.342i)13-s + (0.0703 + 0.153i)14-s + (0.696 − 0.447i)15-s + (0.147 + 0.0433i)16-s + (0.108 + 0.751i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.682771 + 1.78033i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.682771 + 1.78033i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.52 + 1.62i)T \) |
| 23 | \( 1 + (-106. + 27.8i)T \) |
| good | 2 | \( 1 + (-2.95 - 3.41i)T + (-1.13 + 7.91i)T^{2} \) |
| 5 | \( 1 + (6.65 - 14.5i)T + (-81.8 - 94.4i)T^{2} \) |
| 7 | \( 1 + (-1.88 - 0.553i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (16.3 - 18.9i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-54.6 + 16.0i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-7.57 - 52.6i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-16.8 + 117. i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-1.34 - 9.36i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (37.4 - 24.0i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (-172. - 378. i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-111. + 244. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (341. + 219. i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 + 349.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (43.0 + 12.6i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-721. + 211. i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (302. - 194. i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (236. + 272. i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-51.6 - 59.5i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-124. + 867. i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-1.03e3 + 303. i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-394. - 863. i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (726. + 466. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (13.1 - 28.8i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
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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
−15.04195473259590791045312493721, −13.64915069158431595803802186337, −12.88499178974245452093762979728, −11.46437136310299784417947598542, −10.52877387191863725469544759587, −8.194214861649501278737876778395, −7.07669117273589325517177445797, −6.38952954128082958410017999147, −4.94969092486899864535871743051, −3.35122473927115424739952609488,
1.11516275460931921448943134536, 3.56948609516318819392838936849, 4.69798856844670966378503431976, 5.72883670828376498058010896342, 8.141385276847387695266040960764, 9.563481967286743514379034868099, 11.01431581383675592476512416417, 11.62932147439970752462593296877, 12.66622025487284298772135692950, 13.35388026374517371824040344527
