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
−10.54184158259341502027106300582, −9.784711940469464762851706161545, −8.825922681053241657683651035914, −8.354246843657630557182741588414, −7.17722710411550234119023950335, −6.28732452322261399713087032799, −5.67742037692225069446264301189, −4.39504086843708811693405223409, −2.75134987335860938595505800062, −1.47693482740581483760301385579,
0.36793278924670751002484833710, 2.18846251420077889485481479126, 3.85162979033631928362127357137, 4.23407932696637492650370971803, 6.12168040318277076631099393543, 6.71293598319016013924277185086, 7.47544300081308213680804385642, 8.814343470156336373521985618155, 9.390073789508955198992300761593, 10.35123004840217256213330072506