| L(s) = 1 | + (0.331 − 0.107i)2-s + (3.76 + 5.17i)3-s + (−6.37 + 4.63i)4-s + (10.0 − 4.96i)5-s + (1.80 + 1.31i)6-s − 14.4i·7-s + (−3.25 + 4.48i)8-s + (−4.31 + 13.2i)9-s + (2.78 − 2.72i)10-s + (−4.11 − 12.6i)11-s + (−47.9 − 15.5i)12-s + (−51.6 − 16.7i)13-s + (−1.55 − 4.78i)14-s + (63.3 + 33.1i)15-s + (18.8 − 58.1i)16-s + (−4.55 + 6.26i)17-s + ⋯ |
| L(s) = 1 | + (0.117 − 0.0381i)2-s + (0.724 + 0.996i)3-s + (−0.796 + 0.578i)4-s + (0.896 − 0.444i)5-s + (0.122 + 0.0893i)6-s − 0.778i·7-s + (−0.143 + 0.198i)8-s + (−0.159 + 0.492i)9-s + (0.0882 − 0.0862i)10-s + (−0.112 − 0.347i)11-s + (−1.15 − 0.374i)12-s + (−1.10 − 0.357i)13-s + (−0.0296 − 0.0913i)14-s + (1.09 + 0.571i)15-s + (0.294 − 0.907i)16-s + (−0.0649 + 0.0894i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.29204 + 0.448948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29204 + 0.448948i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-10.0 + 4.96i)T \) |
| good | 2 | \( 1 + (-0.331 + 0.107i)T + (6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (-3.76 - 5.17i)T + (-8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + 14.4iT - 343T^{2} \) |
| 11 | \( 1 + (4.11 + 12.6i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (51.6 + 16.7i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (4.55 - 6.26i)T + (-1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-123. - 89.5i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (137. - 44.8i)T + (9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (133. - 96.7i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-25.0 - 18.1i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (297. + 96.5i)T + (4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (142. - 437. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 43.4iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-200. - 276. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (140. + 193. i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-182. + 562. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (32.7 + 100. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-304. + 419. i)T + (-9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-352. + 256. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-172. + 56.1i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-377. + 274. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (226. - 311. i)T + (-1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-156. - 480. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (519. + 715. i)T + (-2.82e5 + 8.68e5i)T^{2} \) |
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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
−17.11653156770825279835128032826, −16.14209910738415477418183662976, −14.38467277944818273149325349282, −13.77901706593362627242982377337, −12.39255587542548238063204161367, −10.08839145390456888288326216739, −9.410752358811713271638784678950, −7.947695103323214304148767172694, −5.11552180730180432747525181352, −3.55191702682565951556822852509,
2.24081194872062175494513051546, 5.39757530013828423413131071805, 7.12942210038481056646623015911, 8.926554531386093628397521981924, 9.994335769520833282491975013710, 12.21335831666377981248209246455, 13.54764096044466545802781839649, 14.11719121528089777205581697579, 15.28020231970103091574348166004, 17.49425917512891536255229167133
