L(s) = 1 | + (−1.00 − 1.72i)2-s + (−0.343 − 0.830i)3-s + (−1.97 + 3.47i)4-s + (−1.68 + 4.07i)5-s + (−1.08 + 1.42i)6-s + (−1.87 + 1.87i)7-s + (7.99 − 0.0754i)8-s + (5.79 − 5.79i)9-s + (8.74 − 1.17i)10-s + (1.83 − 4.42i)11-s + (3.56 + 0.446i)12-s + (−5.16 − 12.4i)13-s + (5.11 + 1.35i)14-s + 3.96·15-s + (−8.17 − 13.7i)16-s − 25.5i·17-s + ⋯ |
L(s) = 1 | + (−0.502 − 0.864i)2-s + (−0.114 − 0.276i)3-s + (−0.494 + 0.869i)4-s + (−0.337 + 0.815i)5-s + (−0.181 + 0.238i)6-s + (−0.267 + 0.267i)7-s + (0.999 − 0.00943i)8-s + (0.643 − 0.643i)9-s + (0.874 − 0.117i)10-s + (0.166 − 0.402i)11-s + (0.297 + 0.0372i)12-s + (−0.397 − 0.958i)13-s + (0.365 + 0.0966i)14-s + 0.264·15-s + (−0.510 − 0.859i)16-s − 1.50i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.554153 - 0.760623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.554153 - 0.760623i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.00 + 1.72i)T \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
good | 3 | \( 1 + (0.343 + 0.830i)T + (-6.36 + 6.36i)T^{2} \) |
| 5 | \( 1 + (1.68 - 4.07i)T + (-17.6 - 17.6i)T^{2} \) |
| 11 | \( 1 + (-1.83 + 4.42i)T + (-85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + (5.16 + 12.4i)T + (-119. + 119. i)T^{2} \) |
| 17 | \( 1 + 25.5iT - 289T^{2} \) |
| 19 | \( 1 + (-20.0 + 8.29i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-19.0 - 19.0i)T + 529iT^{2} \) |
| 29 | \( 1 + (28.8 - 11.9i)T + (594. - 594. i)T^{2} \) |
| 31 | \( 1 + 20.2iT - 961T^{2} \) |
| 37 | \( 1 + (-20.6 + 49.7i)T + (-968. - 968. i)T^{2} \) |
| 41 | \( 1 + (-48.4 + 48.4i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (1.98 - 4.80i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + 0.301T + 2.20e3T^{2} \) |
| 53 | \( 1 + (57.5 + 23.8i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-10.1 - 4.18i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-41.4 + 17.1i)T + (2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 + (-20.6 - 49.8i)T + (-3.17e3 + 3.17e3i)T^{2} \) |
| 71 | \( 1 + (61.3 - 61.3i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (-41.1 + 41.1i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 68.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + (105. - 43.7i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-48.8 - 48.8i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + 140.T + 9.40e3T^{2} \) |
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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
−11.51968603207381760932988844701, −11.00409644733020379339838569995, −9.690100293817755003649556146754, −9.200402005050949253352628979998, −7.52649044704755506164601834870, −7.14047230204638793989613108306, −5.37113232703055295570628362185, −3.64364978227610726603125623574, −2.74331056796888542935867375505, −0.70190009597951723264134841132,
1.42567693098736823053473029966, 4.22093091056358563241012139175, 4.91662742439320055960589760297, 6.31833305118617793860549743203, 7.40410587001515296178589191297, 8.281571310056606394904526807656, 9.355708313338311643740551753389, 10.08223291277630275832676525397, 11.11992228679861328893858579735, 12.50052092373987045923725736819