| L(s) = 1 | + (5.17 + 1.51i)2-s + (0.426 − 2.96i)3-s + (17.7 + 11.3i)4-s + (0.676 + 1.48i)5-s + (6.71 − 14.7i)6-s + (12.3 + 3.61i)7-s + (46.0 + 53.1i)8-s + (−8.63 − 2.53i)9-s + (1.24 + 8.68i)10-s + (−13.5 − 29.6i)11-s + (41.3 − 47.7i)12-s + (6.93 − 8.00i)13-s + (58.2 + 37.4i)14-s + (4.68 − 1.37i)15-s + (87.5 + 191. i)16-s + (−56.0 + 35.9i)17-s + ⋯ |
| L(s) = 1 | + (1.82 + 0.536i)2-s + (0.0821 − 0.571i)3-s + (2.21 + 1.42i)4-s + (0.0604 + 0.132i)5-s + (0.457 − 1.00i)6-s + (0.665 + 0.195i)7-s + (2.03 + 2.34i)8-s + (−0.319 − 0.0939i)9-s + (0.0394 + 0.274i)10-s + (−0.371 − 0.813i)11-s + (0.994 − 1.14i)12-s + (0.147 − 0.170i)13-s + (1.11 + 0.714i)14-s + (0.0806 − 0.0236i)15-s + (1.36 + 2.99i)16-s + (−0.799 + 0.513i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.526i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.19977 + 1.48103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.19977 + 1.48103i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.426 + 2.96i)T \) |
| 67 | \( 1 + (484. - 257. i)T \) |
| good | 2 | \( 1 + (-5.17 - 1.51i)T + (6.73 + 4.32i)T^{2} \) |
| 5 | \( 1 + (-0.676 - 1.48i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (-12.3 - 3.61i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (13.5 + 29.6i)T + (-871. + 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-6.93 + 8.00i)T + (-312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (56.0 - 35.9i)T + (2.04e3 - 4.46e3i)T^{2} \) |
| 19 | \( 1 + (16.5 - 4.87i)T + (5.77e3 - 3.70e3i)T^{2} \) |
| 23 | \( 1 + (8.66 - 60.2i)T + (-1.16e4 - 3.42e3i)T^{2} \) |
| 29 | \( 1 - 58.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + (109. + 126. i)T + (-4.23e3 + 2.94e4i)T^{2} \) |
| 37 | \( 1 + 83.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + (135. - 86.7i)T + (2.86e4 - 6.26e4i)T^{2} \) |
| 43 | \( 1 + (-207. + 133. i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 + (-9.51 + 66.1i)T + (-9.96e4 - 2.92e4i)T^{2} \) |
| 53 | \( 1 + (501. + 322. i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (513. + 592. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (370. - 811. i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 71 | \( 1 + (-398. - 256. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (126. - 277. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (-603. + 696. i)T + (-7.01e4 - 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-609. - 1.33e3i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (22.0 + 153. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
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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
−12.41805374158642017595517714536, −11.45400113723592281767028298086, −10.77442766618793346753251092539, −8.535185107030271800187682180949, −7.70935978015517756221305599055, −6.56453779082723575781979307253, −5.76665979275511832074207117857, −4.68851112731522912948651564434, −3.35395902529531903575166479467, −2.09132871054211402504425263202,
1.83030626277814998115813361455, 3.15322862208802775415148458931, 4.57121273117739330677387719259, 4.91034515154666411789176819785, 6.30499612227713902280770784929, 7.47250947847724187279173372754, 9.205077802814893272166153664475, 10.57294364630671552013627075647, 11.01329848037391442747129581255, 12.10507252175612882613497358656
