L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.628 − 0.456i)3-s + (−0.809 − 0.587i)4-s + (−0.766 − 2.10i)5-s + (0.628 − 0.456i)6-s − 4.62·7-s + (0.809 − 0.587i)8-s + (−0.740 − 2.27i)9-s + (2.23 − 0.0798i)10-s + (0.182 − 0.561i)11-s + (0.240 + 0.738i)12-s + (1.06 + 3.28i)13-s + (1.43 − 4.40i)14-s + (−0.477 + 1.66i)15-s + (0.309 + 0.951i)16-s + (6.08 − 4.42i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.362 − 0.263i)3-s + (−0.404 − 0.293i)4-s + (−0.342 − 0.939i)5-s + (0.256 − 0.186i)6-s − 1.74·7-s + (0.286 − 0.207i)8-s + (−0.246 − 0.759i)9-s + (0.706 − 0.0252i)10-s + (0.0549 − 0.169i)11-s + (0.0692 + 0.213i)12-s + (0.295 + 0.909i)13-s + (0.382 − 1.17i)14-s + (−0.123 + 0.431i)15-s + (0.0772 + 0.237i)16-s + (1.47 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.594 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.114252 + 0.226628i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.114252 + 0.226628i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.766 + 2.10i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
good | 3 | \( 1 + (0.628 + 0.456i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 4.62T + 7T^{2} \) |
| 11 | \( 1 + (-0.182 + 0.561i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.06 - 3.28i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-6.08 + 4.42i)T + (5.25 - 16.1i)T^{2} \) |
| 23 | \( 1 + (2.36 - 7.27i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (7.65 + 5.55i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (7.03 - 5.10i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.23 - 3.79i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.136 + 0.420i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 + (-7.46 - 5.42i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.55 + 1.85i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.02 - 6.22i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.74 - 8.44i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (0.436 - 0.317i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (4.90 + 3.56i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.05 - 12.4i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.34 - 3.15i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.322 + 0.234i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.28 + 7.01i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (8.76 + 6.37i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.808125595967506342871749965703, −9.403054435841203387548353272038, −8.911216475036255244567158952294, −7.52125104140008705700770776610, −7.08107581947751785905908555260, −5.78923581049901969404018377072, −5.69913558044471251657657094329, −4.05392985814210471140222380070, −3.29658220068387570628795748386, −1.10105178096932174998532524857,
0.15692481495434240077082716977, 2.33565121440550195827689936795, 3.37663557423372144377312141365, 3.89636736962705738056325928772, 5.58311473315298781930771383373, 6.13712584915996422735730266940, 7.35281752749161825627178944965, 8.012422534439668776957498899371, 9.218827929698535698783790061688, 10.03849248664231863993363228287