L(s) = 1 | + (−1 + 1.73i)2-s + (−1.5 − 2.59i)3-s + (−1.99 − 3.46i)4-s + (−2.29 + 3.97i)5-s + 6·6-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (−4.58 − 7.94i)10-s + (3.24 + 5.61i)11-s + (−6.00 + 10.3i)12-s + 45.2·13-s + 13.7·15-s + (−8 + 13.8i)16-s + (−40.7 − 70.6i)17-s + (−9 − 15.5i)18-s + (2.52 − 4.37i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.205 + 0.355i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.145 − 0.251i)10-s + (0.0888 + 0.153i)11-s + (−0.144 + 0.249i)12-s + 0.964·13-s + 0.236·15-s + (−0.125 + 0.216i)16-s + (−0.581 − 1.00i)17-s + (−0.117 − 0.204i)18-s + (0.0305 − 0.0528i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.03152936850\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03152936850\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (2.29 - 3.97i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-3.24 - 5.61i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 45.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (40.7 + 70.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-2.52 + 4.37i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (53.1 - 92.0i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (146. + 253. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (57.2 - 99.2i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 161.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 471.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (173. - 299. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (202. + 351. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-126. - 219. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-375. + 650. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (5.82 + 10.0i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (342. + 593. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (0.132 - 0.228i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 437.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-29.2 + 50.6i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.28e3T + 9.12e5T^{2} \) |
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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
−11.19987273938446235784657109521, −9.836308422606290416029651860644, −8.953884849781951120537248835094, −7.81948518195331734908697103267, −7.08394673065323396971392043583, −6.14419450477495134192734546752, −5.10669363235352891889385624852, −3.56542141614852808616657614039, −1.72051441086490872413057017539, −0.01379998886448811030699679187,
1.65773691382372707941037784954, 3.44287607904797674857479021146, 4.34670945592688112463290671904, 5.64491628838590276655365555977, 6.86055737871897503984086352813, 8.429034222567772476796946078535, 8.787324443792690220869497665126, 10.07535571818713417708134497908, 10.78013559037680471617233405782, 11.54252403931031919943064327119