L(s) = 1 | + (−1.53 + 1.28i)2-s + (−6.58 − 5.52i)3-s + (0.694 − 3.93i)4-s + (−2.85 + 1.03i)5-s + 17.1·6-s + (−0.561 + 0.204i)7-s + (4.00 + 6.92i)8-s + (8.13 + 46.1i)9-s + (3.03 − 5.26i)10-s + (18.8 + 32.7i)11-s + (−26.3 + 22.0i)12-s + (−5.40 + 30.6i)13-s + (0.598 − 1.03i)14-s + (24.5 + 8.92i)15-s + (−15.0 − 5.47i)16-s + (9.73 + 55.1i)17-s + ⋯ |
L(s) = 1 | + (−0.541 + 0.454i)2-s + (−1.26 − 1.06i)3-s + (0.0868 − 0.492i)4-s + (−0.255 + 0.0929i)5-s + 1.16·6-s + (−0.0303 + 0.0110i)7-s + (0.176 + 0.306i)8-s + (0.301 + 1.70i)9-s + (0.0960 − 0.166i)10-s + (0.517 + 0.896i)11-s + (−0.633 + 0.531i)12-s + (−0.115 + 0.654i)13-s + (0.0114 − 0.0197i)14-s + (0.422 + 0.153i)15-s + (−0.234 − 0.0855i)16-s + (0.138 + 0.787i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00619 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00619 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.312922 + 0.310988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.312922 + 0.310988i\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (223. + 25.7i)T \) |
good | 3 | \( 1 + (6.58 + 5.52i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (2.85 - 1.03i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (0.561 - 0.204i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-18.8 - 32.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (5.40 - 30.6i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-9.73 - 55.1i)T + (-4.61e3 + 1.68e3i)T^{2} \) |
| 19 | \( 1 + (27.5 + 23.0i)T + (1.19e3 + 6.75e3i)T^{2} \) |
| 23 | \( 1 + (-44.7 + 77.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-84.1 - 145. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 248.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (57.4 - 325. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + 526.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (45.4 - 78.7i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (233. + 84.8i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-159. - 58.0i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (14.1 - 80.1i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (16.8 - 6.14i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-877. - 736. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 - 915.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (1.26e3 - 461. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (47.4 + 269. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-164. - 59.7i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-679. + 1.17e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.41806202697600934259760828838, −13.00959869153011059262405781500, −12.05881662949965471496967892825, −11.19079624967284466580821237025, −9.948436135358989427494994689168, −8.309996716974663031102854604992, −6.97803931177938642160408979082, −6.41379488609182365547270307029, −4.84643772730723652579283307481, −1.52008984839685786688010298082,
0.41823879766310764321681212013, 3.58012229941089530085858798830, 5.05193252153273530049081836345, 6.42319945906404546076711693192, 8.268670500586259530569407702941, 9.663694393938127710716299765874, 10.45701567880638509371349568911, 11.53544702663286540465973640934, 12.04832411629395051359506105020, 13.69590286018230384128055808364