L(s) = 1 | + (1.53 + 1.28i)2-s + (4.80 − 4.03i)3-s + (0.694 + 3.93i)4-s + (14.5 + 5.31i)5-s + 12.5·6-s + (−23.2 − 8.47i)7-s + (−4.00 + 6.92i)8-s + (2.14 − 12.1i)9-s + (15.5 + 26.9i)10-s + (23.4 − 40.6i)11-s + (19.2 + 16.1i)12-s + (6.16 + 34.9i)13-s + (−24.7 − 42.9i)14-s + (91.5 − 33.3i)15-s + (−15.0 + 5.47i)16-s + (−5.54 + 31.4i)17-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (0.924 − 0.775i)3-s + (0.0868 + 0.492i)4-s + (1.30 + 0.475i)5-s + 0.853·6-s + (−1.25 − 0.457i)7-s + (−0.176 + 0.306i)8-s + (0.0793 − 0.450i)9-s + (0.491 + 0.850i)10-s + (0.643 − 1.11i)11-s + (0.462 + 0.387i)12-s + (0.131 + 0.746i)13-s + (−0.473 − 0.819i)14-s + (1.57 − 0.573i)15-s + (−0.234 + 0.0855i)16-s + (−0.0791 + 0.448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.65480 + 0.230166i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.65480 + 0.230166i\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (62.5 + 216. i)T \) |
good | 3 | \( 1 + (-4.80 + 4.03i)T + (4.68 - 26.5i)T^{2} \) |
| 5 | \( 1 + (-14.5 - 5.31i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (23.2 + 8.47i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (-23.4 + 40.6i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-6.16 - 34.9i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (5.54 - 31.4i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 19 | \( 1 + (82.3 - 69.0i)T + (1.19e3 - 6.75e3i)T^{2} \) |
| 23 | \( 1 + (90.0 + 156. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-50.8 + 88.0i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 138.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (-61.4 - 348. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + 342.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-177. - 307. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-128. + 46.7i)T + (1.14e5 - 9.56e4i)T^{2} \) |
| 59 | \( 1 + (20.6 - 7.50i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (64.8 + 367. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-411. - 149. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-575. + 483. i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 - 587.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-1.30e3 - 474. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-122. + 693. i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-52.5 + 19.1i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-367. - 636. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.05297437608270219504760085514, −13.34708620688561228198604064075, −12.53919340396078427267314123358, −10.65664135009461238459130467313, −9.373483614837666893906434992092, −8.229088240209047745434322988666, −6.53484818696075412917746224694, −6.25642305622465118617359648393, −3.65452984224317874687363302477, −2.20599631443775439557033835428,
2.24335342343008486386971656391, 3.63877302453910346089480157960, 5.25204132241355992922291654875, 6.58953051262908800176339173568, 8.961376005007379034382139913312, 9.572325099573739104840928300599, 10.24925280545132311260563742347, 12.19545833501946284642746266078, 13.12236143740934862734379439478, 13.88289768052575127783067855572