L(s) = 1 | + (1.38 + 0.268i)2-s + (−1.37 − 1.05i)3-s + (1.85 + 0.744i)4-s + (−0.965 − 0.258i)5-s + (−1.62 − 1.83i)6-s + (−1.22 + 2.13i)7-s + (2.37 + 1.53i)8-s + (0.782 + 2.89i)9-s + (−1.27 − 0.618i)10-s + (−4.86 + 1.30i)11-s + (−1.76 − 2.97i)12-s + (4.41 + 1.18i)13-s + (−2.27 + 2.62i)14-s + (1.05 + 1.37i)15-s + (2.89 + 2.76i)16-s + 2.66i·17-s + ⋯ |
L(s) = 1 | + (0.981 + 0.189i)2-s + (−0.794 − 0.607i)3-s + (0.928 + 0.372i)4-s + (−0.431 − 0.115i)5-s + (−0.664 − 0.747i)6-s + (−0.464 + 0.805i)7-s + (0.840 + 0.541i)8-s + (0.260 + 0.965i)9-s + (−0.402 − 0.195i)10-s + (−1.46 + 0.393i)11-s + (−0.510 − 0.859i)12-s + (1.22 + 0.328i)13-s + (−0.609 + 0.702i)14-s + (0.272 + 0.354i)15-s + (0.722 + 0.691i)16-s + 0.647i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36067 + 1.02383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36067 + 1.02383i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 - 0.268i)T \) |
| 3 | \( 1 + (1.37 + 1.05i)T \) |
| 5 | \( 1 + (0.965 + 0.258i)T \) |
good | 7 | \( 1 + (1.22 - 2.13i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.86 - 1.30i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-4.41 - 1.18i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 - 2.66iT - 17T^{2} \) |
| 19 | \( 1 + (-4.37 - 4.37i)T + 19iT^{2} \) |
| 23 | \( 1 + (1.54 - 0.891i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.05 + 0.818i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-0.0521 + 0.0300i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.47 + 7.47i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.656 + 1.13i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.466 - 1.73i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (5.44 - 9.43i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.08 + 4.08i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.913 + 3.40i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.60 - 13.4i)T + (-52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-2.82 + 10.5i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 16.0iT - 71T^{2} \) |
| 73 | \( 1 + 2.04iT - 73T^{2} \) |
| 79 | \( 1 + (5.10 + 2.94i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.18 + 15.6i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + (2.22 - 3.84i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.86554203621158070419957496740, −10.08952013384513873920326044034, −8.469104843184818502557207988837, −7.78479762524906721678164887479, −6.90744176863657188877624558523, −5.81554442188971582188699851066, −5.53555666019263562882583623630, −4.29279907694568526800271328533, −3.06749389742908394863804676383, −1.77155355363099470105846254600,
0.71542652791104839406192794636, 3.02497487862675580550556923944, 3.68271010319058471135974312140, 4.85373665266399018483265755811, 5.46114874455905191164042788693, 6.56538338198741019072870043632, 7.23833371724071554072454901018, 8.432876020725497193658864129222, 9.928242106827658174923977030392, 10.44105842382434189702224300046