L(s) = 1 | + (−1.23 + 0.686i)2-s − 0.359·3-s + (1.05 − 1.69i)4-s + (0.565 − 2.16i)5-s + (0.443 − 0.246i)6-s − i·7-s + (−0.138 + 2.82i)8-s − 2.87·9-s + (0.786 + 3.06i)10-s − 1.56i·11-s + (−0.379 + 0.609i)12-s − 6.61·13-s + (0.686 + 1.23i)14-s + (−0.203 + 0.776i)15-s + (−1.76 − 3.58i)16-s − 2.81i·17-s + ⋯ |
L(s) = 1 | + (−0.874 + 0.485i)2-s − 0.207·3-s + (0.528 − 0.849i)4-s + (0.253 − 0.967i)5-s + (0.181 − 0.100i)6-s − 0.377i·7-s + (−0.0490 + 0.998i)8-s − 0.957·9-s + (0.248 + 0.968i)10-s − 0.472i·11-s + (−0.109 + 0.176i)12-s − 1.83·13-s + (0.183 + 0.330i)14-s + (−0.0524 + 0.200i)15-s + (−0.442 − 0.896i)16-s − 0.683i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.289352 - 0.394417i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.289352 - 0.394417i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.23 - 0.686i)T \) |
| 5 | \( 1 + (-0.565 + 2.16i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + 0.359T + 3T^{2} \) |
| 11 | \( 1 + 1.56iT - 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 17 | \( 1 + 2.81iT - 17T^{2} \) |
| 19 | \( 1 - 5.37iT - 19T^{2} \) |
| 23 | \( 1 + 5.85iT - 23T^{2} \) |
| 29 | \( 1 + 6.75iT - 29T^{2} \) |
| 31 | \( 1 - 9.18T + 31T^{2} \) |
| 37 | \( 1 - 2.55T + 37T^{2} \) |
| 41 | \( 1 + 7.93T + 41T^{2} \) |
| 43 | \( 1 + 7.16T + 43T^{2} \) |
| 47 | \( 1 + 5.24iT - 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 - 3.87iT - 59T^{2} \) |
| 61 | \( 1 - 4.69iT - 61T^{2} \) |
| 67 | \( 1 - 9.92T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 3.67iT - 73T^{2} \) |
| 79 | \( 1 + 8.46T + 79T^{2} \) |
| 83 | \( 1 - 1.20T + 83T^{2} \) |
| 89 | \( 1 + 4.21T + 89T^{2} \) |
| 97 | \( 1 - 5.89iT - 97T^{2} \) |
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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
−11.68592661538436400180127330877, −10.22396243905656196137340111367, −9.750668110037197813282555418062, −8.537260241157304586470748035186, −7.981373082879824587321359688574, −6.68024416870332240827554880497, −5.61099628285291156302969879746, −4.72822005616781435990908996464, −2.44030171285610330926486777553, −0.46735374875591884519409519487,
2.26046298446017662185611973427, 3.15694981311199833030510939132, 5.07165278106551967933475586249, 6.53297778540016059178144391167, 7.30521862741018520780065951517, 8.409139821551360457985213304942, 9.551471014604798488639735605831, 10.14791799287219964692486225023, 11.22543351691695299184940779132, 11.79393439550846104233610979378