| 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
−13.15883149453803671432228006674, −12.61325453223815958419439629506, −11.60688608862770356582708671485, −10.46602212991142386557285234657, −9.210748451821484303176417886480, −7.00429081218101532008313737209, −5.95153989153661922023264016923, −5.36592471169515006626305324191, −2.52841920085329008149404503858, −0.57284348706783087095014079946,
3.52188053972071093539501935499, 5.05026564594919403959341292403, 6.06352269230455991791648661987, 6.93235372294809687106126995612, 9.625481072122341256370150586477, 10.05492580952778420192106070210, 11.16537781588659398667220828256, 12.65877969912687774178401500468, 13.35126873235359376730127942996, 15.01588309873079746172746206960
