| L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (−0.142 + 0.989i)5-s + (0.841 − 0.540i)6-s + (−1.37 − 3.01i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (4.09 − 1.20i)11-s + (−0.959 + 0.281i)12-s + (−1.47 + 3.21i)13-s + (0.471 + 3.28i)14-s + (−0.654 − 0.755i)15-s + (0.415 + 0.909i)16-s + (−1.63 + 1.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (−0.378 + 0.436i)3-s + (0.420 + 0.270i)4-s + (−0.0636 + 0.442i)5-s + (0.343 − 0.220i)6-s + (−0.520 − 1.13i)7-s + (−0.231 − 0.267i)8-s + (−0.0474 − 0.329i)9-s + (0.131 − 0.287i)10-s + (1.23 − 0.362i)11-s + (−0.276 + 0.0813i)12-s + (−0.407 + 0.892i)13-s + (0.126 + 0.876i)14-s + (−0.169 − 0.195i)15-s + (0.103 + 0.227i)16-s + (−0.395 + 0.254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.327178 - 0.414006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.327178 - 0.414006i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 5 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (4.79 - 0.140i)T \) |
| good | 7 | \( 1 + (1.37 + 3.01i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-4.09 + 1.20i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.47 - 3.21i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (1.63 - 1.04i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (5.63 + 3.62i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-5.55 + 3.57i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (3.67 + 4.24i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (1.32 + 9.21i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.502 + 3.49i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.71 + 3.13i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + (1.98 + 4.33i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-2.90 + 6.36i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.09 - 1.26i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (15.0 + 4.43i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-4.06 - 1.19i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (8.31 + 5.34i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-2.80 + 6.13i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (1.30 + 9.09i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (9.44 - 10.8i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (2.63 - 18.3i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.35123004840217256213330072506, −9.390073789508955198992300761593, −8.814343470156336373521985618155, −7.47544300081308213680804385642, −6.71293598319016013924277185086, −6.12168040318277076631099393543, −4.23407932696637492650370971803, −3.85162979033631928362127357137, −2.18846251420077889485481479126, −0.36793278924670751002484833710,
1.47693482740581483760301385579, 2.75134987335860938595505800062, 4.39504086843708811693405223409, 5.67742037692225069446264301189, 6.28732452322261399713087032799, 7.17722710411550234119023950335, 8.354246843657630557182741588414, 8.825922681053241657683651035914, 9.784711940469464762851706161545, 10.54184158259341502027106300582
