| 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
−13.23610947891733732750764665504, −12.46681155231330428351059364570, −11.02720239502712838533443991456, −10.45629847017379603738031924943, −8.680561559876885915768865294096, −8.293182654482410208446337488627, −6.22830618561016985197923756375, −4.98496144715975945376376028924, −2.64542076389965507161634282070, −0.20821596759025387993326785303,
2.45956337271175161957059430557, 4.90437302910768566620976017019, 6.71065698293004993069748573398, 7.25472797412481903599142074988, 9.014391428868249221540747623038, 10.08846852386758171670943171863, 11.10247579151397039975199993754, 12.16652375144839804775384759166, 13.57293016133591709770302213445, 14.88966483177677314020416087647
