L(s) = 1 | + (1.01 − 0.985i)2-s − 3-s + (0.0573 − 1.99i)4-s + 1.66i·5-s + (−1.01 + 0.985i)6-s − 0.193·7-s + (−1.91 − 2.08i)8-s + 9-s + (1.64 + 1.69i)10-s − 5.43·11-s + (−0.0573 + 1.99i)12-s − 3.51i·13-s + (−0.196 + 0.190i)14-s − 1.66i·15-s + (−3.99 − 0.229i)16-s − 4.63·17-s + ⋯ |
L(s) = 1 | + (0.717 − 0.696i)2-s − 0.577·3-s + (0.0286 − 0.999i)4-s + 0.746i·5-s + (−0.414 + 0.402i)6-s − 0.0731·7-s + (−0.676 − 0.736i)8-s + 0.333·9-s + (0.520 + 0.535i)10-s − 1.63·11-s + (−0.0165 + 0.577i)12-s − 0.975i·13-s + (−0.0524 + 0.0510i)14-s − 0.430i·15-s + (−0.998 − 0.0573i)16-s − 1.12·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0438479 + 0.698500i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0438479 + 0.698500i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.01 + 0.985i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (0.790 - 8.14i)T \) |
good | 5 | \( 1 - 1.66iT - 5T^{2} \) |
| 7 | \( 1 + 0.193T + 7T^{2} \) |
| 11 | \( 1 + 5.43T + 11T^{2} \) |
| 13 | \( 1 + 3.51iT - 13T^{2} \) |
| 17 | \( 1 + 4.63T + 17T^{2} \) |
| 19 | \( 1 + 1.14iT - 19T^{2} \) |
| 23 | \( 1 + 7.82iT - 23T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 + 8.23T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 0.746iT - 41T^{2} \) |
| 43 | \( 1 + 3.05T + 43T^{2} \) |
| 47 | \( 1 - 2.28iT - 47T^{2} \) |
| 53 | \( 1 - 6.58iT - 53T^{2} \) |
| 59 | \( 1 + 1.92iT - 59T^{2} \) |
| 61 | \( 1 - 1.81iT - 61T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 - 5.05T + 73T^{2} \) |
| 79 | \( 1 - 8.43T + 79T^{2} \) |
| 83 | \( 1 - 8.12iT - 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
| 97 | \( 1 + 8.73iT - 97T^{2} \) |
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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
−10.30679415821408558082472759226, −9.290961858255623400476794616957, −8.000646885969651410015421346846, −6.96005478201978937434592829044, −6.10744267930457843856575533486, −5.25472590526171154512820218876, −4.42774642304216480413988130310, −3.06152112011141986985370379020, −2.31883425424630559862041107346, −0.26991393292343422336989653963,
2.12121593870486800544132675360, 3.64713851652245961323138826768, 4.74703294400633446648541915703, 5.28481463207772592793115684477, 6.17682531675347027523421349921, 7.20441297415273002413061846845, 7.900630046556476629314575117402, 8.896743075108490047588836856072, 9.691470577048294126639475127815, 11.08528389871207457114806980311