| L(s) = 1 | + (2.05 + 1.43i)2-s + (0.551 − 1.64i)3-s + (1.46 + 4.03i)4-s + (−1.82 + 1.29i)5-s + (3.49 − 2.58i)6-s + (−1.14 − 2.44i)7-s + (−1.48 + 5.55i)8-s + (−2.39 − 1.81i)9-s + (−5.60 + 0.0236i)10-s + (−2.09 − 2.50i)11-s + (7.43 − 0.187i)12-s + (1.80 + 2.57i)13-s + (1.17 − 6.67i)14-s + (1.11 + 3.70i)15-s + (−4.47 + 3.75i)16-s + (1.57 + 5.88i)17-s + ⋯ |
| L(s) = 1 | + (1.45 + 1.01i)2-s + (0.318 − 0.948i)3-s + (0.734 + 2.01i)4-s + (−0.816 + 0.577i)5-s + (1.42 − 1.05i)6-s + (−0.431 − 0.925i)7-s + (−0.526 + 1.96i)8-s + (−0.797 − 0.603i)9-s + (−1.77 + 0.00747i)10-s + (−0.633 − 0.754i)11-s + (2.14 − 0.0541i)12-s + (0.500 + 0.714i)13-s + (0.314 − 1.78i)14-s + (0.287 + 0.957i)15-s + (−1.11 + 0.939i)16-s + (0.382 + 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88373 + 0.773059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88373 + 0.773059i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.551 + 1.64i)T \) |
| 5 | \( 1 + (1.82 - 1.29i)T \) |
| good | 2 | \( 1 + (-2.05 - 1.43i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (1.14 + 2.44i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (2.09 + 2.50i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.80 - 2.57i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.57 - 5.88i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.91 + 1.68i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.50 + 2.10i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.541 - 3.07i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.346 + 0.125i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (4.28 - 1.14i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-7.54 - 1.33i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.386 + 4.42i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (6.23 - 2.90i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-3.48 - 3.48i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.87 - 4.92i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (6.73 + 2.45i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.315 - 0.221i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (8.46 + 4.88i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.0248 + 0.00667i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-10.9 + 1.93i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.627 - 0.896i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (3.93 + 6.81i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.10 + 0.446i)T + (95.5 + 16.8i)T^{2} \) |
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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
−13.62172781465252344486380156425, −12.71432212115965304496370894144, −11.83636055398266865083212441669, −10.69245210748240359384670246298, −8.407331528100623860885293367530, −7.60173052699069162339620701352, −6.75914477858195906751241090043, −5.92186588135242449452654500799, −4.04630032221599036120744888663, −3.16939997532496262675332587300,
2.68811933424497643968320830832, 3.72767379261413293737320558766, 4.94934871826406499694391839335, 5.64568271396777328941043473720, 7.88640775551537077390773040360, 9.363676085857083323417295568989, 10.22290201149043995276555145091, 11.45720066716243456467376687683, 12.07246321437597227842559679825, 12.97561575516961313275971816288
