| L(s) = 1 | + (−1.87 + 0.684i)2-s + (−1.99 − 0.725i)3-s + (3.06 − 2.57i)4-s + (1.44 + 8.17i)5-s + 4.24·6-s + (−1.26 − 7.18i)7-s + (−4.00 + 6.92i)8-s + (−17.2 − 14.4i)9-s + (−8.30 − 14.3i)10-s + (−32.0 + 55.4i)11-s + (−7.96 + 2.90i)12-s + (−38.7 + 32.5i)13-s + (7.29 + 12.6i)14-s + (3.05 − 17.3i)15-s + (2.77 − 15.7i)16-s + (−75.3 − 63.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (−0.383 − 0.139i)3-s + (0.383 − 0.321i)4-s + (0.128 + 0.731i)5-s + 0.288·6-s + (−0.0683 − 0.387i)7-s + (−0.176 + 0.306i)8-s + (−0.638 − 0.535i)9-s + (−0.262 − 0.454i)10-s + (−0.877 + 1.52i)11-s + (−0.191 + 0.0697i)12-s + (−0.827 + 0.694i)13-s + (0.139 + 0.241i)14-s + (0.0526 − 0.298i)15-s + (0.0434 − 0.246i)16-s + (−1.07 − 0.901i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0338559 + 0.270095i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0338559 + 0.270095i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.87 - 0.684i)T \) |
| 37 | \( 1 + (196. - 109. i)T \) |
| good | 3 | \( 1 + (1.99 + 0.725i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (-1.44 - 8.17i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (1.26 + 7.18i)T + (-322. + 117. i)T^{2} \) |
| 11 | \( 1 + (32.0 - 55.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (38.7 - 32.5i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (75.3 + 63.2i)T + (853. + 4.83e3i)T^{2} \) |
| 19 | \( 1 + (93.0 + 33.8i)T + (5.25e3 + 4.40e3i)T^{2} \) |
| 23 | \( 1 + (-92.9 - 160. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-16.8 + 29.2i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 120.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (-249. + 209. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + 434.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (29.1 + 50.5i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (9.57 - 54.3i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (121. - 687. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (-58.2 + 48.8i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (114. + 651. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-492. - 179. i)T + (2.74e5 + 2.30e5i)T^{2} \) |
| 73 | \( 1 + 616.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-82.0 - 465. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-486. - 408. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-93.8 + 532. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-5.20 - 9.00i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.88966483177677314020416087647, −13.57293016133591709770302213445, −12.16652375144839804775384759166, −11.10247579151397039975199993754, −10.08846852386758171670943171863, −9.014391428868249221540747623038, −7.25472797412481903599142074988, −6.71065698293004993069748573398, −4.90437302910768566620976017019, −2.45956337271175161957059430557,
0.20821596759025387993326785303, 2.64542076389965507161634282070, 4.98496144715975945376376028924, 6.22830618561016985197923756375, 8.293182654482410208446337488627, 8.680561559876885915768865294096, 10.45629847017379603738031924943, 11.02720239502712838533443991456, 12.46681155231330428351059364570, 13.23610947891733732750764665504
