L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.18 − 1.26i)3-s + (−0.499 + 0.866i)4-s + (−0.686 + 1.18i)5-s + (0.5 − 1.65i)6-s − 0.999·8-s + (−0.186 + 2.99i)9-s − 1.37·10-s + (−2.18 − 3.78i)11-s + (1.68 − 0.396i)12-s + (1 − 1.73i)13-s + (2.31 − 0.543i)15-s + (−0.5 − 0.866i)16-s + 4.37·17-s + (−2.68 + 1.33i)18-s − 5·19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.684 − 0.728i)3-s + (−0.249 + 0.433i)4-s + (−0.306 + 0.531i)5-s + (0.204 − 0.677i)6-s − 0.353·8-s + (−0.0620 + 0.998i)9-s − 0.433·10-s + (−0.659 − 1.14i)11-s + (0.486 − 0.114i)12-s + (0.277 − 0.480i)13-s + (0.597 − 0.140i)15-s + (−0.125 − 0.216i)16-s + 1.06·17-s + (−0.633 + 0.314i)18-s − 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.905018 - 0.456734i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.905018 - 0.456734i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (1.18 + 1.26i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.686 - 1.18i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.18 + 3.78i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + (-3.68 + 6.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.37 + 2.37i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-5.18 + 8.98i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.55 + 7.89i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.74T + 53T^{2} \) |
| 59 | \( 1 + (-3.55 + 6.16i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.74 + 4.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 + (-4.55 - 7.89i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.39464900582480740732662937104, −8.780066708954714976301906405130, −8.083817375210178970275345099377, −7.36624592029444471228049046457, −6.48032582706566435465766279699, −5.79303530703679504942693763688, −5.00612751876087593455304802864, −3.64162141721823896375200343178, −2.54110089053881379362304815798, −0.51851934664225093502716716900,
1.32693635089189741031910185607, 2.97829142051715479256209401247, 4.17162404354524650813944691647, 4.75819663681857561496220244630, 5.57744561892566941631190842050, 6.61813138476618736180225682753, 7.77132788625453368588953569702, 8.889824971172755546992227980777, 9.675620035054704321902157602579, 10.29850023019607044120775905202