| L(s) = 1 | + (−1.53 − 1.28i)2-s + (−1.34 + 1.12i)3-s + (0.694 + 3.93i)4-s + (6.03 + 2.19i)5-s + 3.50·6-s + (−28.8 − 10.4i)7-s + (4.00 − 6.92i)8-s + (−4.15 + 23.5i)9-s + (−6.42 − 11.1i)10-s + (−18.0 + 31.3i)11-s + (−5.36 − 4.50i)12-s + (10.9 + 62.0i)13-s + (30.6 + 53.1i)14-s + (−10.5 + 3.84i)15-s + (−15.0 + 5.47i)16-s + (−1.92 + 10.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.258 + 0.216i)3-s + (0.0868 + 0.492i)4-s + (0.539 + 0.196i)5-s + 0.238·6-s + (−1.55 − 0.566i)7-s + (0.176 − 0.306i)8-s + (−0.153 + 0.873i)9-s + (−0.203 − 0.351i)10-s + (−0.495 + 0.858i)11-s + (−0.129 − 0.108i)12-s + (0.233 + 1.32i)13-s + (0.585 + 1.01i)14-s + (−0.181 + 0.0662i)15-s + (−0.234 + 0.0855i)16-s + (−0.0275 + 0.156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.287894 + 0.427816i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.287894 + 0.427816i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-198. + 106. i)T \) |
| good | 3 | \( 1 + (1.34 - 1.12i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (-6.03 - 2.19i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (28.8 + 10.4i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (18.0 - 31.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-10.9 - 62.0i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (1.92 - 10.9i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (29.3 - 24.6i)T + (1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (55.9 + 96.9i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (135. - 234. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 209.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (77.3 + 438. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 - 347.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-49.0 - 84.9i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (201. - 73.2i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (333. - 121. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-124. - 706. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (384. + 139. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (362. - 304. i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 - 269.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (366. + 133. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (85.2 - 483. i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-907. + 330. i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-492. - 852. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.15607799947229920338728276392, −13.20412514059506075312456472533, −12.22859537394097298672743138941, −10.67737666431851724178276835424, −10.11127323631756802169069010448, −9.075518567499701141925856567015, −7.36827231611001890899346093470, −6.20033721251327843975042076552, −4.19817358654477076988889015523, −2.31541587570016459154720749121,
0.38409854742092100513090534956, 3.07745898557674619832198329644, 5.84017171178025207868071330554, 6.19836045571021410507201840055, 7.951334397161008103677998558631, 9.307317815955778801063115712231, 9.968003697351916421649955135991, 11.48945040113603953221694057586, 12.86523899086332432269914236571, 13.50162946887465853078129877319
