| L(s) = 1 | + (0.0741 + 0.0476i)2-s + (−0.142 − 0.989i)3-s + (−0.827 − 1.81i)4-s + (−1.48 − 0.435i)5-s + (0.0366 − 0.0801i)6-s + (−0.654 + 0.755i)7-s + (0.0500 − 0.348i)8-s + (−0.959 + 0.281i)9-s + (−0.0892 − 0.103i)10-s + (0.0405 − 0.0260i)11-s + (−1.67 + 1.07i)12-s + (−1.24 − 1.43i)13-s + (−0.0845 + 0.0248i)14-s + (−0.220 + 1.53i)15-s + (−2.58 + 2.98i)16-s + (−0.415 + 0.909i)17-s + ⋯ |
| L(s) = 1 | + (0.0524 + 0.0337i)2-s + (−0.0821 − 0.571i)3-s + (−0.413 − 0.906i)4-s + (−0.663 − 0.194i)5-s + (0.0149 − 0.0327i)6-s + (−0.247 + 0.285i)7-s + (0.0177 − 0.123i)8-s + (−0.319 + 0.0939i)9-s + (−0.0282 − 0.0325i)10-s + (0.0122 − 0.00784i)11-s + (−0.483 + 0.310i)12-s + (−0.345 − 0.399i)13-s + (−0.0226 + 0.00663i)14-s + (−0.0568 + 0.395i)15-s + (−0.647 + 0.746i)16-s + (−0.100 + 0.220i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0598307 + 0.337268i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0598307 + 0.337268i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (3.50 - 3.27i)T \) |
| good | 2 | \( 1 + (-0.0741 - 0.0476i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (1.48 + 0.435i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-0.0405 + 0.0260i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.24 + 1.43i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.415 - 0.909i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.953 + 2.08i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.80 - 8.32i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.284 + 1.97i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (2.34 - 0.689i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.26 - 0.372i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.525 + 3.65i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 0.701T + 47T^{2} \) |
| 53 | \( 1 + (-7.68 + 8.86i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (5.26 + 6.07i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-2.12 + 14.8i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (9.85 + 6.33i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-4.02 - 2.58i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (4.64 + 10.1i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (0.288 + 0.333i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (1.68 - 0.494i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (0.146 + 1.01i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (6.26 + 1.83i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.52597102572370892663311419246, −9.556745179541839711923446473048, −8.700530263295549254576780396217, −7.76129023255132994709763100768, −6.72866893474516269441524572285, −5.75264333276471368464113161265, −4.88033334766991123162639271991, −3.59700998680346101039660612547, −1.89625115020314564616712845060, −0.20061794217677125029438509540,
2.68004793404254326345360866856, 3.94938279943696394117865270019, 4.34331663479970415363061473765, 5.81229754111763425609898419969, 7.13625746979776799467760278664, 7.87306813996378320547086924345, 8.788922172332344940442866724193, 9.676425281187761036842224177244, 10.57539913213265995928661091394, 11.69547148421132948210338596532
