| L(s) = 1 | + (−1.30 + 1.51i)2-s + (−2.52 + 1.62i)3-s + (−0.569 − 3.95i)4-s + (6.04 + 13.2i)5-s + (0.853 − 5.93i)6-s + (29.2 − 8.59i)7-s + (6.73 + 4.32i)8-s + (3.73 − 8.18i)9-s + (−27.9 − 8.19i)10-s + (−31.1 − 35.9i)11-s + (7.85 + 9.06i)12-s + (68.3 + 20.0i)13-s + (−25.3 + 55.5i)14-s + (−36.7 − 23.5i)15-s + (−15.3 + 4.50i)16-s + (−18.4 + 128. i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.540 + 1.18i)5-s + (0.0580 − 0.404i)6-s + (1.58 − 0.464i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.882 − 0.259i)10-s + (−0.853 − 0.985i)11-s + (0.189 + 0.218i)12-s + (1.45 + 0.428i)13-s + (−0.483 + 1.05i)14-s + (−0.632 − 0.406i)15-s + (−0.239 + 0.0704i)16-s + (−0.262 + 1.82i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.880223 + 1.04059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.880223 + 1.04059i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 - 1.51i)T \) |
| 3 | \( 1 + (2.52 - 1.62i)T \) |
| 23 | \( 1 + (-3.92 - 110. i)T \) |
| good | 5 | \( 1 + (-6.04 - 13.2i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (-29.2 + 8.59i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (31.1 + 35.9i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-68.3 - 20.0i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (18.4 - 128. i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-6.80 - 47.3i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-18.3 + 127. i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (148. + 95.2i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (84.2 - 184. i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-0.354 - 0.775i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (53.3 - 34.3i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 - 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (320. - 94.1i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-437. - 128. i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (159. + 102. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-672. + 775. i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (330. - 381. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-111. - 775. i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (557. + 163. i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-456. + 998. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-4.70 + 3.02i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-274. - 600. i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−13.38649133160921787356542502856, −11.31907508104016999162869380405, −10.90283195366784727071847720698, −10.22505299151948535927306306318, −8.572525177511849542771297844810, −7.75959198407064604121408256405, −6.31588854791455156290763524090, −5.59186898934784181260469105100, −3.87835715476908739736808823260, −1.60489669173144546518009625804,
0.959513353263904827948790920583, 2.17865275709380060448708605451, 4.77649259120259238646606927909, 5.33761819905815632292205920865, 7.28739351114549166568713001584, 8.452662032849570415732000741890, 9.132031344323130633981539183382, 10.59454568628125289892113339119, 11.37650429264398275335062571020, 12.38591107999244162416804691031
