L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.993 − 1.41i)3-s + (−0.499 + 0.866i)4-s + (0.246 − 0.142i)5-s + (0.731 − 1.56i)6-s + (−2.09 + 1.61i)7-s − 0.999·8-s + (−1.02 + 2.81i)9-s + (0.246 + 0.142i)10-s + (−2.18 − 2.49i)11-s + (1.72 − 0.151i)12-s + 6.02i·13-s + (−2.44 − 1.00i)14-s + (−0.446 − 0.207i)15-s + (−0.5 − 0.866i)16-s + (−3.12 + 5.41i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.573 − 0.819i)3-s + (−0.249 + 0.433i)4-s + (0.110 − 0.0635i)5-s + (0.298 − 0.640i)6-s + (−0.791 + 0.611i)7-s − 0.353·8-s + (−0.341 + 0.939i)9-s + (0.0778 + 0.0449i)10-s + (−0.658 − 0.752i)11-s + (0.498 − 0.0436i)12-s + 1.67i·13-s + (−0.654 − 0.268i)14-s + (−0.115 − 0.0536i)15-s + (−0.125 − 0.216i)16-s + (−0.758 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.229223 + 0.634173i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.229223 + 0.634173i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.993 + 1.41i)T \) |
| 7 | \( 1 + (2.09 - 1.61i)T \) |
| 11 | \( 1 + (2.18 + 2.49i)T \) |
good | 5 | \( 1 + (-0.246 + 0.142i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 6.02iT - 13T^{2} \) |
| 17 | \( 1 + (3.12 - 5.41i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.70 + 2.71i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.55 - 0.898i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.434T + 29T^{2} \) |
| 31 | \( 1 + (3.64 - 6.31i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.57 + 9.65i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.326T + 41T^{2} \) |
| 43 | \( 1 - 2.29iT - 43T^{2} \) |
| 47 | \( 1 + (3.52 - 2.03i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.66 + 0.960i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.38 - 4.83i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.37 - 1.95i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.34 + 7.51i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.88iT - 71T^{2} \) |
| 73 | \( 1 + (-3.80 - 2.19i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.63 + 2.67i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.61T + 83T^{2} \) |
| 89 | \( 1 + (-14.8 + 8.56i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.98T + 97T^{2} \) |
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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
−11.57462790593555633161075135971, −10.75483438168468365658293783797, −9.306963131634121791055434601447, −8.641233042314813669023978413106, −7.46127252911368968160109249881, −6.64566720412921859157768266827, −5.91279477213148874774192285051, −5.08601831031001064696737107218, −3.56326048593207223430611027678, −2.03487069686538197187700205088,
0.37865472353905925514018055159, 2.76767852234272687953769080881, 3.73121350452179990880582166653, 4.89862062608306671739208824340, 5.64267343017724987550111251650, 6.80024723161633355716223300337, 7.986484904787376656142252301419, 9.500803058287538929118289338198, 10.00843352528878550361090443967, 10.52906237188697630477508082666