| L(s) = 1 | + (0.536 − 1.65i)2-s + (−2.42 − 1.76i)3-s + (4.03 + 2.92i)4-s + (8.44 + 7.32i)5-s + (−4.21 + 3.06i)6-s + 7.66·7-s + (18.2 − 13.2i)8-s + (2.78 + 8.55i)9-s + (16.6 − 10.0i)10-s + (6.32 − 19.4i)11-s + (−4.61 − 14.2i)12-s + (−4.57 − 14.0i)13-s + (4.11 − 12.6i)14-s + (−7.58 − 32.6i)15-s + (0.205 + 0.633i)16-s + (88.1 − 64.0i)17-s + ⋯ |
| L(s) = 1 | + (0.189 − 0.584i)2-s + (−0.467 − 0.339i)3-s + (0.503 + 0.366i)4-s + (0.755 + 0.655i)5-s + (−0.286 + 0.208i)6-s + 0.413·7-s + (0.806 − 0.585i)8-s + (0.103 + 0.317i)9-s + (0.526 − 0.316i)10-s + (0.173 − 0.533i)11-s + (−0.111 − 0.341i)12-s + (−0.0976 − 0.300i)13-s + (0.0785 − 0.241i)14-s + (−0.130 − 0.562i)15-s + (0.00321 + 0.00990i)16-s + (1.25 − 0.914i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.521i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.852 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.81777 - 0.512031i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81777 - 0.512031i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.42 + 1.76i)T \) |
| 5 | \( 1 + (-8.44 - 7.32i)T \) |
| good | 2 | \( 1 + (-0.536 + 1.65i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 - 7.66T + 343T^{2} \) |
| 11 | \( 1 + (-6.32 + 19.4i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (4.57 + 14.0i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-88.1 + 64.0i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (79.1 - 57.4i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (24.8 - 76.6i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (47.7 + 34.7i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (142. - 103. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (71.2 + 219. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-55.4 - 170. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (386. + 280. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-417. - 303. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (176. + 543. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (16.1 - 49.7i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-446. + 324. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (298. + 216. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (125. - 386. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-529. - 384. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (341. - 247. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-131. + 404. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (477. + 347. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−13.76513176119202246963116246397, −12.71484640528904382216997991461, −11.67584148415104392356963205942, −10.84899281478456285460355822122, −9.861123198649793758885761756466, −7.927307381734666454384289149036, −6.78380145590467467724182332604, −5.49024083431257161210617934380, −3.34875190113635617023782562799, −1.75089328746350227835567821503,
1.72496131987287211321492757778, 4.61511200884204902239227003059, 5.66192209143142240144432957754, 6.73236378038674553810103154048, 8.297978445413948796561346294163, 9.785573956784698726564708135245, 10.70456455864306587924650782557, 11.96114229213703796462987920173, 13.11598802913124314406397857521, 14.49505165692390889103513775565
