| L(s) = 1 | + (1.61 − 1.17i)2-s + (−2.47 − 7.62i)3-s + (1.23 − 3.80i)4-s + (−4.04 − 2.93i)5-s + (−12.9 − 9.42i)6-s + (−0.0702 + 0.216i)7-s + (−2.47 − 7.60i)8-s + (−30.1 + 21.9i)9-s − 10·10-s + (29.2 − 21.8i)11-s − 32.0·12-s + (−54.0 + 39.2i)13-s + (0.140 + 0.432i)14-s + (−12.3 + 38.1i)15-s + (−12.9 − 9.40i)16-s + (43.3 + 31.5i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.476 − 1.46i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (−0.882 − 0.641i)6-s + (−0.00379 + 0.0116i)7-s + (−0.109 − 0.336i)8-s + (−1.11 + 0.811i)9-s − 0.316·10-s + (0.800 − 0.598i)11-s − 0.771·12-s + (−1.15 + 0.838i)13-s + (0.00268 + 0.00825i)14-s + (−0.213 + 0.656i)15-s + (−0.202 − 0.146i)16-s + (0.618 + 0.449i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00706860 + 1.42284i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00706860 + 1.42284i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 + 1.17i)T \) |
| 5 | \( 1 + (4.04 + 2.93i)T \) |
| 11 | \( 1 + (-29.2 + 21.8i)T \) |
| good | 3 | \( 1 + (2.47 + 7.62i)T + (-21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 + (0.0702 - 0.216i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (54.0 - 39.2i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-43.3 - 31.5i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (28.7 + 88.4i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-59.0 + 181. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-224. + 163. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-60.4 + 186. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (106. + 329. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 54.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-154. - 475. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (83.6 - 60.7i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (9.10 - 28.0i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (390. + 283. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 672.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-682. - 495. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-200. + 615. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (95.7 - 69.5i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (45.1 + 32.7i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 175.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (393. - 285. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−12.39637090365894815288116363945, −11.98705827290267195737948369419, −11.11048876405454563108710776581, −9.465586341965991674644539528738, −7.978689589369644680694419040468, −6.84558207601142838680268725681, −5.90854272526272810467181067378, −4.31294670501503769172230942944, −2.28746253816755879794791754789, −0.69471539368664497666577835752,
3.27765174572234263038444753738, 4.46592693374000834379522556433, 5.37961657515542072015639486474, 6.81740207514986461391312092262, 8.224843811637650957301596811672, 9.783116878017684298607885536855, 10.38227097146452669751497402121, 11.77713034063479890066812828811, 12.37705889436889484508161517855, 14.14419282101270061524694755529
