| L(s) = 1 | + (0.359 + 2.27i)2-s + (−0.930 + 1.46i)3-s + (−3.12 + 1.01i)4-s + (−0.136 − 0.0697i)5-s + (−3.65 − 1.58i)6-s + (1.38 − 0.333i)7-s + (−1.34 − 2.63i)8-s + (−1.26 − 2.71i)9-s + (0.109 − 0.335i)10-s + (0.659 − 0.772i)11-s + (1.42 − 5.51i)12-s + (2.99 + 4.88i)13-s + (1.25 + 3.03i)14-s + (0.229 − 0.135i)15-s + (0.189 − 0.137i)16-s + (−4.60 + 0.362i)17-s + ⋯ |
| L(s) = 1 | + (0.254 + 1.60i)2-s + (−0.537 + 0.843i)3-s + (−1.56 + 0.507i)4-s + (−0.0612 − 0.0311i)5-s + (−1.49 − 0.648i)6-s + (0.524 − 0.125i)7-s + (−0.475 − 0.932i)8-s + (−0.422 − 0.906i)9-s + (0.0345 − 0.106i)10-s + (0.198 − 0.232i)11-s + (0.411 − 1.59i)12-s + (0.830 + 1.35i)13-s + (0.335 + 0.810i)14-s + (0.0591 − 0.0348i)15-s + (0.0472 − 0.0343i)16-s + (−1.11 + 0.0879i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0686218 + 0.989311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0686218 + 0.989311i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.930 - 1.46i)T \) |
| 41 | \( 1 + (-0.530 + 6.38i)T \) |
| good | 2 | \( 1 + (-0.359 - 2.27i)T + (-1.90 + 0.618i)T^{2} \) |
| 5 | \( 1 + (0.136 + 0.0697i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (-1.38 + 0.333i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (-0.659 + 0.772i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.99 - 4.88i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (4.60 - 0.362i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (-2.40 + 3.91i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-5.59 - 4.06i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.0606 + 0.00477i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (-4.62 - 1.50i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.402 - 1.23i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-8.28 + 1.31i)T + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.46 + 6.08i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-0.764 + 9.70i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (0.765 - 1.05i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.108 - 0.686i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (6.60 + 7.73i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (10.1 + 8.65i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (9.51 - 9.51i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.107 - 0.258i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 5.23iT - 83T^{2} \) |
| 89 | \( 1 + (-3.19 - 13.2i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (2.64 - 2.25i)T + (15.1 - 95.8i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.11977669085234043164210929566, −13.46227966392554060077152628790, −11.68581828985095585522176546372, −10.95402269713292096860945276268, −9.252230418169748148448198422028, −8.618166796997970212846109669326, −7.06923363564127047884174845606, −6.19605358990519450315312060203, −4.96726313733417598219376998728, −4.09312092455479970298979659159,
1.26280266660233618301676365403, 2.84847198584670463857533518407, 4.58759256850787093857142017288, 5.94052259539291302522500249106, 7.59880394530293719584311114053, 8.863599772468054353973452096124, 10.36309632485159341773709528956, 11.11419657166063461447880825792, 11.81042087917193712799613501294, 12.87025513036057624405943558644
