L(s) = 1 | + (−1 − i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (−4.93 + 0.829i)5-s + 2.44·6-s + (1.87 + 1.87i)7-s + (2 − 2i)8-s − 2.99i·9-s + (5.76 + 4.10i)10-s − 5.74·11-s + (−2.44 − 2.44i)12-s + (15.0 − 15.0i)13-s − 3.74i·14-s + (5.02 − 7.05i)15-s − 4·16-s + (−4.78 − 4.78i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (−0.986 + 0.165i)5-s + 0.408·6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.576 + 0.410i)10-s − 0.522·11-s + (−0.204 − 0.204i)12-s + (1.15 − 1.15i)13-s − 0.267i·14-s + (0.334 − 0.470i)15-s − 0.250·16-s + (−0.281 − 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0650 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0650 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.441135 - 0.470822i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.441135 - 0.470822i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 + (4.93 - 0.829i)T \) |
| 7 | \( 1 + (-1.87 - 1.87i)T \) |
good | 11 | \( 1 + 5.74T + 121T^{2} \) |
| 13 | \( 1 + (-15.0 + 15.0i)T - 169iT^{2} \) |
| 17 | \( 1 + (4.78 + 4.78i)T + 289iT^{2} \) |
| 19 | \( 1 + 17.4iT - 361T^{2} \) |
| 23 | \( 1 + (-13.2 + 13.2i)T - 529iT^{2} \) |
| 29 | \( 1 + 37.7iT - 841T^{2} \) |
| 31 | \( 1 + 27.0T + 961T^{2} \) |
| 37 | \( 1 + (-11.3 - 11.3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 53.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-37.1 + 37.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (9.39 + 9.39i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-43.8 + 43.8i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 62.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 1.67T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-28.4 - 28.4i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 47.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (31.2 - 31.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 107. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-18.2 + 18.2i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 174. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (91.5 + 91.5i)T + 9.40e3iT^{2} \) |
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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
−11.53718809086597319503572295690, −11.01227221308770591904640815131, −10.20545685997947001282759899834, −8.833415901931957212152722920183, −8.123573982670183968654440850327, −6.94725039151420426532538086699, −5.41814022951615400073468133446, −4.11453576898355198152536297033, −2.87179214096906934885447081707, −0.49606625727679552315257615614,
1.39950008238396848672532526061, 3.81590513560130877485141928478, 5.14448648271434780789362562651, 6.45781358156630169478216757405, 7.39702038495747657862796580660, 8.257839607878554625140527077275, 9.167265601581869092391595716378, 10.73171995264262040128294357648, 11.22130887202472713763661588377, 12.29365342708379248612425682045