| 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
−11.69547148421132948210338596532, −10.57539913213265995928661091394, −9.676425281187761036842224177244, −8.788922172332344940442866724193, −7.87306813996378320547086924345, −7.13625746979776799467760278664, −5.81229754111763425609898419969, −4.34331663479970415363061473765, −3.94938279943696394117865270019, −2.68004793404254326345360866856,
0.20061794217677125029438509540, 1.89625115020314564616712845060, 3.59700998680346101039660612547, 4.88033334766991123162639271991, 5.75264333276471368464113161265, 6.72866893474516269441524572285, 7.76129023255132994709763100768, 8.700530263295549254576780396217, 9.556745179541839711923446473048, 10.52597102572370892663311419246
