L(s) = 1 | + (1.53 + 1.28i)2-s + (−7.21 + 6.05i)3-s + (0.694 + 3.93i)4-s + (11.5 + 4.20i)5-s − 18.8·6-s + (−26.9 − 9.80i)7-s + (−4.00 + 6.92i)8-s + (10.7 − 60.8i)9-s + (12.3 + 21.3i)10-s + (−21.0 + 36.4i)11-s + (−28.8 − 24.2i)12-s + (−2.03 − 11.5i)13-s + (−28.6 − 49.6i)14-s + (−108. + 39.6i)15-s + (−15.0 + 5.47i)16-s + (−17.2 + 97.6i)17-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (−1.38 + 1.16i)3-s + (0.0868 + 0.492i)4-s + (1.03 + 0.376i)5-s − 1.28·6-s + (−1.45 − 0.529i)7-s + (−0.176 + 0.306i)8-s + (0.397 − 2.25i)9-s + (0.389 + 0.674i)10-s + (−0.577 + 0.999i)11-s + (−0.694 − 0.582i)12-s + (−0.0434 − 0.246i)13-s + (−0.547 − 0.948i)14-s + (−1.87 + 0.682i)15-s + (−0.234 + 0.0855i)16-s + (−0.245 + 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0547i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0253623 - 0.925716i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0253623 - 0.925716i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.53 - 1.28i)T \) |
| 37 | \( 1 + (-36.5 - 222. i)T \) |
good | 3 | \( 1 + (7.21 - 6.05i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (-11.5 - 4.20i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (26.9 + 9.80i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (21.0 - 36.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (2.03 + 11.5i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (17.2 - 97.6i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (-55.5 + 46.6i)T + (1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (-90.5 - 156. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (22.5 - 39.1i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 92.9T + 2.97e4T^{2} \) |
| 41 | \( 1 + (-56.4 - 320. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 - 21.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + (135. + 233. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-71.7 + 26.1i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-736. + 268. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (90.2 + 511. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (15.0 + 5.47i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-325. + 273. i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + 36.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-246. - 89.6i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (133. - 754. i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-161. + 58.6i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (173. + 301. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−15.01588309873079746172746206960, −13.35126873235359376730127942996, −12.65877969912687774178401500468, −11.16537781588659398667220828256, −10.05492580952778420192106070210, −9.625481072122341256370150586477, −6.93235372294809687106126995612, −6.06352269230455991791648661987, −5.05026564594919403959341292403, −3.52188053972071093539501935499,
0.57284348706783087095014079946, 2.52841920085329008149404503858, 5.36592471169515006626305324191, 5.95153989153661922023264016923, 7.00429081218101532008313737209, 9.210748451821484303176417886480, 10.46602212991142386557285234657, 11.60688608862770356582708671485, 12.61325453223815958419439629506, 13.15883149453803671432228006674