L(s) = 1 | + (−2.51 − 1.45i)2-s + (−2.26 + 4.67i)3-s + (0.214 + 0.371i)4-s + (−2.5 + 4.33i)5-s + (12.4 − 8.47i)6-s + (17.9 + 4.37i)7-s + 21.9i·8-s + (−16.7 − 21.1i)9-s + (12.5 − 7.25i)10-s + (−28.8 + 16.6i)11-s + (−2.22 + 0.162i)12-s − 62.4i·13-s + (−38.8 − 37.1i)14-s + (−14.5 − 21.4i)15-s + (33.6 − 58.2i)16-s + (−34.7 − 60.1i)17-s + ⋯ |
L(s) = 1 | + (−0.888 − 0.513i)2-s + (−0.435 + 0.900i)3-s + (0.0268 + 0.0464i)4-s + (−0.223 + 0.387i)5-s + (0.849 − 0.576i)6-s + (0.971 + 0.236i)7-s + 0.971i·8-s + (−0.620 − 0.783i)9-s + (0.397 − 0.229i)10-s + (−0.790 + 0.456i)11-s + (−0.0534 + 0.00391i)12-s − 1.33i·13-s + (−0.742 − 0.708i)14-s + (−0.251 − 0.369i)15-s + (0.525 − 0.909i)16-s + (−0.495 − 0.858i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0923104 - 0.232787i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0923104 - 0.232787i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.26 - 4.67i)T \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
| 7 | \( 1 + (-17.9 - 4.37i)T \) |
good | 2 | \( 1 + (2.51 + 1.45i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (28.8 - 16.6i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 62.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (34.7 + 60.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (71.6 + 41.3i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (49.8 + 28.7i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 114. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (187. - 108. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-136. + 236. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 36.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 185.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (25.2 - 43.6i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (204. - 117. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (267. + 462. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (313. + 181. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-232. - 403. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.07e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (771. - 445. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (636. - 1.10e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 245.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (548. - 950. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.08e3iT - 9.12e5T^{2} \) |
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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
−12.51740778531019121251290818506, −11.17038450403888344966047729611, −10.83357892030694735569377426400, −9.856255862004576037172876780286, −8.748854056794905739217315409744, −7.70072655431833262854830855026, −5.65391358549506619092284563205, −4.64112735210822511308892674033, −2.54956092641370461768825149640, −0.19757049420474136176786141101,
1.62024547505654674259490559091, 4.35760882315371278044048749847, 6.03729909113665177209151342967, 7.29511639772359692350442651989, 8.136003674962866659456674409348, 8.867510175892692953179094422995, 10.55513945790759100836162999892, 11.53375006677665951462640210092, 12.66691842689945498774649024766, 13.50872144153657832442243797167