L(s) = 1 | − 7.20i·2-s + (8.77 + 2.01i)3-s − 35.9·4-s + 11.1i·5-s + (14.5 − 63.2i)6-s + 23.3·7-s + 144. i·8-s + (72.8 + 35.3i)9-s + 80.6·10-s − 33.9i·11-s + (−315. − 72.5i)12-s − 125.·13-s − 168. i·14-s + (−22.5 + 98.0i)15-s + 463.·16-s + 308. i·17-s + ⋯ |
L(s) = 1 | − 1.80i·2-s + (0.974 + 0.224i)3-s − 2.24·4-s + 0.447i·5-s + (0.403 − 1.75i)6-s + 0.476·7-s + 2.25i·8-s + (0.899 + 0.436i)9-s + 0.806·10-s − 0.280i·11-s + (−2.19 − 0.503i)12-s − 0.745·13-s − 0.858i·14-s + (−0.100 + 0.435i)15-s + 1.80·16-s + 1.06i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.843993 - 1.06010i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.843993 - 1.06010i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-8.77 - 2.01i)T \) |
| 5 | \( 1 - 11.1iT \) |
good | 2 | \( 1 + 7.20iT - 16T^{2} \) |
| 7 | \( 1 - 23.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + 33.9iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 125.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 308. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 363.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 102. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.56e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 326.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 72.4T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.14e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.50e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.95e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.96e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 5.04e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 736.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.25e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.30e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.76e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.36e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 6.89e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.40e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.87e3T + 8.85e7T^{2} \) |
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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
−19.02360978357746591755202550050, −17.49104715848186168241008250092, −14.98433454781507841320993160168, −13.83614204849502010616700702928, −12.54913437188157948880901370100, −10.94255400681828369255585473219, −9.830130373738813280646287040215, −8.314438726321527638568650041886, −4.12723968844093698482789691866, −2.31137887005096632444979580234,
4.78395746241869307537520082550, 7.01388605599170706138399106500, 8.234456067188329212862238679530, 9.428111401023613053448076948622, 12.76998990084292776508444807808, 14.10535274401709028493605713906, 14.94108742207748997698544402298, 16.13395925392414341930110194390, 17.44491016516588968870837324696, 18.56949804782934623750602522410