| L(s) = 1 | + (0.907 − 0.419i)2-s + (0.947 + 0.319i)3-s + (0.647 − 0.762i)4-s + (0.401 + 0.241i)5-s + (0.994 − 0.108i)6-s + (−0.0924 + 1.70i)7-s + (0.267 − 0.963i)8-s + (0.796 + 0.605i)9-s + (0.465 + 0.0506i)10-s + (−0.0193 + 0.118i)11-s + (0.856 − 0.515i)12-s + (4.72 − 3.59i)13-s + (0.631 + 1.58i)14-s + (0.303 + 0.357i)15-s + (−0.161 − 0.986i)16-s + (0.104 + 1.92i)17-s + ⋯ |
| L(s) = 1 | + (0.641 − 0.296i)2-s + (0.547 + 0.184i)3-s + (0.323 − 0.381i)4-s + (0.179 + 0.108i)5-s + (0.405 − 0.0441i)6-s + (−0.0349 + 0.644i)7-s + (0.0945 − 0.340i)8-s + (0.265 + 0.201i)9-s + (0.147 + 0.0160i)10-s + (−0.00583 + 0.0355i)11-s + (0.247 − 0.148i)12-s + (1.31 − 0.996i)13-s + (0.168 + 0.423i)14-s + (0.0783 + 0.0922i)15-s + (−0.0404 − 0.246i)16-s + (0.0252 + 0.466i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.31671 - 0.189168i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31671 - 0.189168i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.907 + 0.419i)T \) |
| 3 | \( 1 + (-0.947 - 0.319i)T \) |
| 59 | \( 1 + (7.09 - 2.95i)T \) |
| good | 5 | \( 1 + (-0.401 - 0.241i)T + (2.34 + 4.41i)T^{2} \) |
| 7 | \( 1 + (0.0924 - 1.70i)T + (-6.95 - 0.756i)T^{2} \) |
| 11 | \( 1 + (0.0193 - 0.118i)T + (-10.4 - 3.51i)T^{2} \) |
| 13 | \( 1 + (-4.72 + 3.59i)T + (3.47 - 12.5i)T^{2} \) |
| 17 | \( 1 + (-0.104 - 1.92i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (2.59 - 2.45i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (1.36 - 0.300i)T + (20.8 - 9.65i)T^{2} \) |
| 29 | \( 1 + (7.23 + 3.34i)T + (18.7 + 22.1i)T^{2} \) |
| 31 | \( 1 + (2.74 + 2.59i)T + (1.67 + 30.9i)T^{2} \) |
| 37 | \( 1 + (-2.72 - 9.81i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (3.58 + 0.789i)T + (37.2 + 17.2i)T^{2} \) |
| 43 | \( 1 + (0.304 + 1.85i)T + (-40.7 + 13.7i)T^{2} \) |
| 47 | \( 1 + (0.307 - 0.185i)T + (22.0 - 41.5i)T^{2} \) |
| 53 | \( 1 + (7.66 - 0.833i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (-7.17 + 3.31i)T + (39.4 - 46.4i)T^{2} \) |
| 67 | \( 1 + (0.655 - 2.36i)T + (-57.4 - 34.5i)T^{2} \) |
| 71 | \( 1 + (1.13 - 0.681i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (3.67 + 9.22i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (-1.42 + 0.480i)T + (62.8 - 47.8i)T^{2} \) |
| 83 | \( 1 + (5.80 - 8.56i)T + (-30.7 - 77.1i)T^{2} \) |
| 89 | \( 1 + (5.62 + 2.60i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (-2.38 + 5.99i)T + (-70.4 - 66.7i)T^{2} \) |
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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
−11.46408195118560656851478716998, −10.56106202773853404572126217256, −9.765579647277181349925814799903, −8.608337170973631441057059119950, −7.85587597429261553676839123577, −6.27421704652994664560570266974, −5.64249360605143564179518318569, −4.17895518379850882914326462395, −3.20502043707321776446701062589, −1.90343297900057176318161277154,
1.80436160204553479712266359581, 3.45783041079349365077796429283, 4.29965790015311677453925036699, 5.67699251658551915032006454883, 6.76836387410715969143465725939, 7.50608440556734812163067724839, 8.700153397585628614386357488141, 9.419557252333967074546564177524, 10.83016164380943609828323040569, 11.45098042864353876461746009441
