L(s) = 1 | + (−0.112 − 0.0302i)2-s + (−2.25 + 1.29i)3-s + (−1.72 − 0.993i)4-s + (−1.24 − 1.24i)5-s + (0.293 − 0.0785i)6-s + (−2.18 + 1.49i)7-s + (0.329 + 0.329i)8-s + (1.87 − 3.25i)9-s + (0.103 + 0.178i)10-s + (−0.506 + 1.89i)11-s + 5.16·12-s + (−1.85 − 3.09i)13-s + (0.291 − 0.102i)14-s + (4.43 + 1.18i)15-s + (1.95 + 3.39i)16-s + (−2.13 + 3.70i)17-s + ⋯ |
L(s) = 1 | + (−0.0797 − 0.0213i)2-s + (−1.29 + 0.750i)3-s + (−0.860 − 0.496i)4-s + (−0.558 − 0.558i)5-s + (0.119 − 0.0320i)6-s + (−0.824 + 0.565i)7-s + (0.116 + 0.116i)8-s + (0.625 − 1.08i)9-s + (0.0325 + 0.0564i)10-s + (−0.152 + 0.570i)11-s + 1.49·12-s + (−0.515 − 0.857i)13-s + (0.0778 − 0.0275i)14-s + (1.14 + 0.306i)15-s + (0.489 + 0.848i)16-s + (−0.518 + 0.898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00123326 - 0.0353182i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00123326 - 0.0353182i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.18 - 1.49i)T \) |
| 13 | \( 1 + (1.85 + 3.09i)T \) |
good | 2 | \( 1 + (0.112 + 0.0302i)T + (1.73 + i)T^{2} \) |
| 3 | \( 1 + (2.25 - 1.29i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.24 + 1.24i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.506 - 1.89i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (2.13 - 3.70i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.12 + 1.10i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (5.53 - 3.19i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.57 + 6.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.02 + 3.02i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.732 - 2.73i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (2.94 - 11.0i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (1.55 + 0.896i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.68 - 4.68i)T - 47iT^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 + (0.436 + 1.62i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.66 - 1.54i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.0190 + 0.00510i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.23 - 4.59i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.698 + 0.698i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.93T + 79T^{2} \) |
| 83 | \( 1 + (-9.87 - 9.87i)T + 83iT^{2} \) |
| 89 | \( 1 + (7.76 + 2.07i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-14.2 + 3.82i)T + (84.0 - 48.5i)T^{2} \) |
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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
−15.06297599993898813790524470895, −13.28883932508923685931319061928, −12.41509216778766429382524952269, −11.45260132350960212206523077206, −10.06334664925066855293372507377, −9.630505854871371891236252141258, −8.104887248074270336211672847819, −6.07937944219011133221049449403, −5.19701777486391296568328249832, −4.09986853610558038249842788853,
0.05083097366308138595948967712, 3.64181158332693580763834498962, 5.26970707873687033402887584390, 6.82329186014411233572439745503, 7.45551111067986559680335188635, 9.155687019775643967428133926740, 10.49893907789993754756888064991, 11.63599832054258260409702821859, 12.36205601473699206978505667366, 13.40222234411206017482013776963