L(s) = 1 | + (−1.87 − 0.684i)2-s + (0.520 + 2.95i)3-s + (3.06 + 2.57i)4-s + (16.3 − 13.7i)5-s + (1.04 − 5.90i)6-s + (−6.08 + 10.5i)7-s + (−4.00 − 6.92i)8-s + (−8.45 + 3.07i)9-s + (−40.2 + 14.6i)10-s + (1.05 + 1.83i)11-s + (−6.00 + 10.3i)12-s + (10.9 − 62.3i)13-s + (18.6 − 15.6i)14-s + (49.1 + 41.2i)15-s + (2.77 + 15.7i)16-s + (83.9 + 30.5i)17-s + ⋯ |
L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.100 + 0.568i)3-s + (0.383 + 0.321i)4-s + (1.46 − 1.22i)5-s + (0.0708 − 0.402i)6-s + (−0.328 + 0.569i)7-s + (−0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (−1.27 + 0.462i)10-s + (0.0290 + 0.0502i)11-s + (−0.144 + 0.250i)12-s + (0.234 − 1.32i)13-s + (0.355 − 0.298i)14-s + (0.846 + 0.710i)15-s + (0.0434 + 0.246i)16-s + (1.19 + 0.435i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.48831 - 0.315241i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48831 - 0.315241i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.87 + 0.684i)T \) |
| 3 | \( 1 + (-0.520 - 2.95i)T \) |
| 19 | \( 1 + (-81.9 + 11.9i)T \) |
good | 5 | \( 1 + (-16.3 + 13.7i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (6.08 - 10.5i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-1.05 - 1.83i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-10.9 + 62.3i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-83.9 - 30.5i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-57.4 - 48.2i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-36.6 + 13.3i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (7.81 - 13.5i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 98.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-6.55 - 37.1i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (296. - 248. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-276. + 100. i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + (-97.0 - 81.4i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (538. + 195. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (562. + 472. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (779. - 283. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-340. + 285. i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + (-197. - 1.11e3i)T + (-3.65e5 + 1.33e5i)T^{2} \) |
| 79 | \( 1 + (215. + 1.22e3i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (363. - 628. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (20.3 - 115. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (1.01e3 + 370. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−12.90353590086803024578464517046, −12.11959364941839936513941164196, −10.49443627017403603569174467170, −9.722610176686083315515251368082, −9.049713552704330188023477622754, −8.009858239972815074210020278636, −5.94668965146271388829850914684, −5.18481819827936241079208228221, −3.00578414075128094742081324263, −1.22822249945304653490391551145,
1.54462573773181508691420225049, 3.02820642080051744315865180988, 5.68221426350017298151014767792, 6.73483395070304794291444099738, 7.33796709426623359776535684290, 9.092380816083932674509960926464, 9.932337222723878657572404359494, 10.78768650594064716250208993088, 12.00635859790825282007240746571, 13.75043871606385627687887059890