L(s) = 1 | + (0.0173 − 1.41i)2-s + 3.07i·3-s + (−1.99 − 0.0489i)4-s − i·5-s + (4.35 + 0.0532i)6-s − 1.19·7-s + (−0.103 + 2.82i)8-s − 6.47·9-s + (−1.41 − 0.0173i)10-s − 4.94i·11-s + (0.150 − 6.15i)12-s − 3.51i·13-s + (−0.0206 + 1.68i)14-s + 3.07·15-s + (3.99 + 0.195i)16-s − 0.0467·17-s + ⋯ |
L(s) = 1 | + (0.0122 − 0.999i)2-s + 1.77i·3-s + (−0.999 − 0.0244i)4-s − 0.447i·5-s + (1.77 + 0.0217i)6-s − 0.451·7-s + (−0.0367 + 0.999i)8-s − 2.15·9-s + (−0.447 − 0.00547i)10-s − 1.49i·11-s + (0.0435 − 1.77i)12-s − 0.974i·13-s + (−0.00552 + 0.451i)14-s + 0.794·15-s + (0.998 + 0.0489i)16-s − 0.0113·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0367 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0367 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.642039 - 0.666056i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.642039 - 0.666056i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0173 + 1.41i)T \) |
| 5 | \( 1 + iT \) |
| 19 | \( 1 + iT \) |
good | 3 | \( 1 - 3.07iT - 3T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 + 4.94iT - 11T^{2} \) |
| 13 | \( 1 + 3.51iT - 13T^{2} \) |
| 17 | \( 1 + 0.0467T + 17T^{2} \) |
| 23 | \( 1 - 4.46T + 23T^{2} \) |
| 29 | \( 1 + 7.32iT - 29T^{2} \) |
| 31 | \( 1 - 7.66T + 31T^{2} \) |
| 37 | \( 1 - 9.66iT - 37T^{2} \) |
| 41 | \( 1 + 6.53T + 41T^{2} \) |
| 43 | \( 1 + 5.24iT - 43T^{2} \) |
| 47 | \( 1 - 8.03T + 47T^{2} \) |
| 53 | \( 1 + 5.27iT - 53T^{2} \) |
| 59 | \( 1 + 10.4iT - 59T^{2} \) |
| 61 | \( 1 + 1.73iT - 61T^{2} \) |
| 67 | \( 1 + 12.2iT - 67T^{2} \) |
| 71 | \( 1 + 6.86T + 71T^{2} \) |
| 73 | \( 1 + 8.68T + 73T^{2} \) |
| 79 | \( 1 - 2.95T + 79T^{2} \) |
| 83 | \( 1 - 5.02iT - 83T^{2} \) |
| 89 | \( 1 + 8.38T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−10.02753790735151156692841279784, −9.656935400176107723854456482611, −8.566397593982200795356688106605, −8.316836741363642134943282660982, −6.08318283919525267006600432078, −5.25535039990154682495338489455, −4.50835548821320376353642056550, −3.41778485537637587945874320880, −2.93756334116838364182411227879, −0.50191258617025555794150124882,
1.43652304601652726748147987938, 2.80230030000725460325788945095, 4.36855407452758460026332270537, 5.62137414917852588269901302634, 6.59186244900661916939116590429, 7.05288209464240874210618115028, 7.50757957613350757721363348069, 8.612566148500499251970290371387, 9.357657802131430014525678477903, 10.43821744496858070288244673545