L(s) = 1 | + (−3.95 − 2.28i)2-s + (−1.04 − 5.08i)3-s + (6.41 + 11.1i)4-s + (2.5 − 4.33i)5-s + (−7.46 + 22.5i)6-s + (−15.3 − 10.3i)7-s − 22.0i·8-s + (−24.7 + 10.6i)9-s + (−19.7 + 11.4i)10-s + (−38.8 + 22.4i)11-s + (49.8 − 44.3i)12-s − 21.9i·13-s + (37.1 + 75.9i)14-s + (−24.6 − 8.17i)15-s + (1.01 − 1.75i)16-s + (18.3 + 31.7i)17-s + ⋯ |
L(s) = 1 | + (−1.39 − 0.806i)2-s + (−0.201 − 0.979i)3-s + (0.801 + 1.38i)4-s + (0.223 − 0.387i)5-s + (−0.507 + 1.53i)6-s + (−0.830 − 0.557i)7-s − 0.974i·8-s + (−0.918 + 0.395i)9-s + (−0.624 + 0.360i)10-s + (−1.06 + 0.614i)11-s + (1.19 − 1.06i)12-s − 0.468i·13-s + (0.709 + 1.44i)14-s + (−0.424 − 0.140i)15-s + (0.0157 − 0.0273i)16-s + (0.261 + 0.452i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0278334 + 0.0162301i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0278334 + 0.0162301i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.04 + 5.08i)T \) |
| 5 | \( 1 + (-2.5 + 4.33i)T \) |
| 7 | \( 1 + (15.3 + 10.3i)T \) |
good | 2 | \( 1 + (3.95 + 2.28i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (38.8 - 22.4i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 21.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-18.3 - 31.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-91.7 - 52.9i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-21.5 - 12.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 31.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (262. - 151. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-130. + 226. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 294.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (59.6 - 103. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (560. - 323. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (3.12 + 5.40i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (702. + 405. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-13.4 - 23.2i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 639. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (619. - 357. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (312. - 541. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 630.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-350. + 607. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 528. iT - 9.12e5T^{2} \) |
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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
−12.83422916141150750503713905977, −12.51342518378142994210827833995, −11.11784268916844974596060787926, −10.22514023336446078521362959386, −9.291101344083605654705780925476, −7.953793474247575325532471919403, −7.30350367328796389525177992677, −5.61100755553324266918931094336, −2.97573421361738428219951410734, −1.42860016710315390506601453929,
0.02947620676601402410925422762, 3.05779560198133095146697118958, 5.37374847965282953540092078843, 6.37419209151222071532135650225, 7.70478504947879980267902027962, 9.045047704104220770909284835419, 9.591322517925814195048155780154, 10.53863319986763955884906138368, 11.52944383082782833430677675459, 13.27254588160520772259935106089