L(s) = 1 | + (1.80 + 1.04i)2-s + (1.17 + 2.03i)4-s + (3.53 + 2.04i)7-s + 0.734i·8-s + (−0.675 + 1.17i)11-s + (0.561 − 0.324i)13-s + (4.26 + 7.38i)14-s + (1.58 − 2.74i)16-s − 1.35i·17-s − 0.648·19-s + (−2.44 + 1.41i)22-s + (−4.14 + 2.39i)23-s + 1.35·26-s + 9.61i·28-s + (−1.93 + 3.35i)29-s + ⋯ |
L(s) = 1 | + (1.27 + 0.737i)2-s + (0.587 + 1.01i)4-s + (1.33 + 0.772i)7-s + 0.259i·8-s + (−0.203 + 0.353i)11-s + (0.155 − 0.0898i)13-s + (1.13 + 1.97i)14-s + (0.396 − 0.686i)16-s − 0.327i·17-s − 0.148·19-s + (−0.520 + 0.300i)22-s + (−0.864 + 0.499i)23-s + 0.265·26-s + 1.81i·28-s + (−0.359 + 0.623i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.66375 + 1.84916i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.66375 + 1.84916i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-1.80 - 1.04i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-3.53 - 2.04i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.675 - 1.17i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.561 + 0.324i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.35iT - 17T^{2} \) |
| 19 | \( 1 + 0.648T + 19T^{2} \) |
| 23 | \( 1 + (4.14 - 2.39i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.93 - 3.35i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.84 - 6.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.52iT - 37T^{2} \) |
| 41 | \( 1 + (0.0898 + 0.155i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.710 - 0.410i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (9.44 + 5.45i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4.17iT - 53T^{2} \) |
| 59 | \( 1 + (2.08 + 3.61i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.91 + 3.30i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.05 - 4.07i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.11T + 71T^{2} \) |
| 73 | \( 1 + 12.3iT - 73T^{2} \) |
| 79 | \( 1 + (-5.17 + 8.95i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (10.6 + 6.12i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + (11.7 + 6.79i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.92008922099546550026015486344, −9.791924305962516009009206164211, −8.649483040111415978855599694593, −7.84689429929031203225207070686, −6.99876040425755911473278537828, −5.91999909895473999486953134907, −5.17288408961574865803990569822, −4.55109843741998751251727364777, −3.33133273454539873376821740886, −1.90537150648926827379441457099,
1.45380840350762062427866848379, 2.63918021333472390230592662525, 4.00076726723336806698638423085, 4.49997241040816136379007701497, 5.50494711451438173156029627645, 6.45209743747772355337647248423, 7.889763056840845107618626948935, 8.321383144611295800370336052410, 9.868376813780100243994603858888, 10.72364439712270638211942298636