| L(s) = 1 | + (−0.642 − 1.26i)2-s + (−0.711 − 4.49i)3-s + (−1.17 + 1.61i)4-s + (−4.53 + 2.09i)5-s + (−5.20 + 3.78i)6-s + (3.58 − 3.58i)7-s + (2.79 + 0.442i)8-s + (−11.1 + 3.61i)9-s + (5.55 + 4.37i)10-s + (3.53 − 10.8i)11-s + (8.10 + 4.13i)12-s + (10.0 − 19.7i)13-s + (−6.82 − 2.21i)14-s + (12.6 + 18.9i)15-s + (−1.23 − 3.80i)16-s + (−3.10 + 19.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 − 0.630i)2-s + (−0.237 − 1.49i)3-s + (−0.293 + 0.404i)4-s + (−0.907 + 0.419i)5-s + (−0.867 + 0.630i)6-s + (0.512 − 0.512i)7-s + (0.349 + 0.0553i)8-s + (−1.23 + 0.401i)9-s + (0.555 + 0.437i)10-s + (0.321 − 0.989i)11-s + (0.675 + 0.344i)12-s + (0.773 − 1.51i)13-s + (−0.487 − 0.158i)14-s + (0.843 + 1.26i)15-s + (−0.0772 − 0.237i)16-s + (−0.182 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.249445 - 0.745475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.249445 - 0.745475i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 + 1.26i)T \) |
| 5 | \( 1 + (4.53 - 2.09i)T \) |
| good | 3 | \( 1 + (0.711 + 4.49i)T + (-8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-3.58 + 3.58i)T - 49iT^{2} \) |
| 11 | \( 1 + (-3.53 + 10.8i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (-10.0 + 19.7i)T + (-99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (3.10 - 19.6i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-15.8 - 21.8i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (0.476 - 0.242i)T + (310. - 427. i)T^{2} \) |
| 29 | \( 1 + (3.67 - 5.05i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-5.42 + 3.94i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (5.24 + 2.67i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-7.33 - 22.5i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (-44.7 - 44.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (27.2 - 4.31i)T + (2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (13.6 + 86.0i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-20.7 + 6.73i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (21.3 - 65.6i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-14.4 + 91.1i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-12.7 - 9.24i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (60.4 - 30.7i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-71.8 + 98.9i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-60.8 - 9.64i)T + (6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-75.7 - 24.6i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (134. - 21.2i)T + (8.94e3 - 2.90e3i)T^{2} \) |
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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
−14.57518799382311608172325798992, −13.39678402011865640045259922541, −12.47035643847004205004082917941, −11.41223532445130241149038277525, −10.61683867488715054925939286167, −8.209341600039143547229632478284, −7.77365905387927536185134460235, −6.12488182381568994408454842890, −3.47205549079042952999339578717, −1.04903744021631734678930036286,
4.22349041380162787760010430807, 5.08127952181633903395797701009, 7.13762036709866316128835957010, 8.879717833372188668545171996688, 9.441120102865773994544837373426, 11.10723217902819438108494902782, 11.88438266894182917649077006446, 13.97671221097126509070425985606, 15.18050109405053610888131464059, 15.78401540218891785948093649334
