L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.73 + 0.0789i)3-s + (−0.499 + 0.866i)4-s − 0.460·5-s + (−0.796 − 1.53i)6-s + (2.25 − 1.38i)7-s + 0.999·8-s + (2.98 + 0.273i)9-s + (0.230 + 0.398i)10-s − 3.64·11-s + (−0.933 + 1.45i)12-s + (0.730 + 1.26i)13-s + (−2.32 − 1.26i)14-s + (−0.796 − 0.0363i)15-s + (−0.5 − 0.866i)16-s + (−1.86 − 3.23i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.998 + 0.0455i)3-s + (−0.249 + 0.433i)4-s − 0.205·5-s + (−0.325 − 0.627i)6-s + (0.853 − 0.521i)7-s + 0.353·8-s + (0.995 + 0.0910i)9-s + (0.0728 + 0.126i)10-s − 1.09·11-s + (−0.269 + 0.421i)12-s + (0.202 + 0.350i)13-s + (−0.621 − 0.338i)14-s + (−0.205 − 0.00938i)15-s + (−0.125 − 0.216i)16-s + (−0.452 − 0.784i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10964 - 0.405525i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10964 - 0.405525i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.73 - 0.0789i)T \) |
| 7 | \( 1 + (-2.25 + 1.38i)T \) |
good | 5 | \( 1 + 0.460T + 5T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 + (-0.730 - 1.26i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.86 + 3.23i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.02 - 3.51i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 1.13T + 23T^{2} \) |
| 29 | \( 1 + (4.48 - 7.77i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.257 + 0.445i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.55 - 7.88i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.472 + 0.819i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.66 + 8.07i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.16 + 2.01i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.21 - 10.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.44 + 11.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.04 + 10.4i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.16 + 2.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 + (6.62 + 11.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.32 + 5.75i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.36 - 2.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.59 - 9.68i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.38655916309380363200747855137, −12.24460914882675857656628456312, −10.99498002796629771214023868897, −10.21082096093572138453728571349, −8.985711004925294823556470709152, −8.060286130157985991655445873554, −7.23632067042690379541667706123, −4.88397518256203805059061476386, −3.58701670280910971974432199198, −1.97976735009963365560703596403,
2.28707844821845826024212305852, 4.28222483882954674924873337418, 5.72258534851055428004219357605, 7.37866297649901416844393703707, 8.164884061090475522182973982347, 8.915079072079775931538614957847, 10.17087262508644338245013821218, 11.26641607727892585447047360100, 12.84135577497650400002283873482, 13.57372873928981216857805813547