| L(s) = 1 | + (−2.52 + 0.917i)2-s + (0.900 + 0.519i)3-s + (3.98 − 3.34i)4-s + (0.360 − 0.168i)5-s + (−2.74 − 0.484i)6-s + (1.15 + 4.30i)7-s + (−4.30 + 7.44i)8-s + (−0.959 − 1.66i)9-s + (−0.755 + 0.755i)10-s + (1.40 + 3.01i)11-s + (5.32 − 0.939i)12-s + (1.79 − 1.25i)13-s + (−6.86 − 9.80i)14-s + (0.412 + 0.0360i)15-s + (2.20 − 12.4i)16-s + (−0.652 − 0.174i)17-s + ⋯ |
| L(s) = 1 | + (−1.78 + 0.649i)2-s + (0.519 + 0.300i)3-s + (1.99 − 1.67i)4-s + (0.161 − 0.0752i)5-s + (−1.12 − 0.197i)6-s + (0.436 + 1.62i)7-s + (−1.52 + 2.63i)8-s + (−0.319 − 0.554i)9-s + (−0.238 + 0.238i)10-s + (0.423 + 0.909i)11-s + (1.53 − 0.271i)12-s + (0.498 − 0.348i)13-s + (−1.83 − 2.62i)14-s + (0.106 + 0.00930i)15-s + (0.550 − 3.12i)16-s + (−0.158 − 0.0424i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.442630 + 0.310447i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.442630 + 0.310447i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 73 | \( 1 + (-1.49 - 8.41i)T \) |
| good | 2 | \( 1 + (2.52 - 0.917i)T + (1.53 - 1.28i)T^{2} \) |
| 3 | \( 1 + (-0.900 - 0.519i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.360 + 0.168i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-1.15 - 4.30i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.40 - 3.01i)T + (-7.07 + 8.42i)T^{2} \) |
| 13 | \( 1 + (-1.79 + 1.25i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (0.652 + 0.174i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.11 + 3.06i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-2.00 + 5.50i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.84 - 0.859i)T + (18.6 + 22.2i)T^{2} \) |
| 31 | \( 1 + (0.277 + 3.17i)T + (-30.5 + 5.38i)T^{2} \) |
| 37 | \( 1 + (7.28 + 2.65i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (1.75 + 9.95i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.32 - 0.355i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (1.29 - 1.85i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-0.602 + 1.29i)T + (-34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (-4.24 - 6.06i)T + (-20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (-4.95 + 0.873i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.87 - 0.506i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-9.93 + 3.61i)T + (54.3 - 45.6i)T^{2} \) |
| 79 | \( 1 + (5.30 + 0.936i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (11.7 - 11.7i)T - 83iT^{2} \) |
| 89 | \( 1 + (-0.548 + 3.11i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (1.59 - 0.922i)T + (48.5 - 84.0i)T^{2} \) |
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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
−15.28065297077896275018200374939, −14.48588652481702239486332238221, −12.26122855238104022750661722057, −11.17528268365315912774254099296, −9.784508453826075634580743988708, −8.908395330838337906288915843581, −8.456651042294023222531844162736, −6.84516690658110829851664126646, −5.58827075804058364233400972966, −2.27454318175477742764542079375,
1.50713262326508640124728135691, 3.49552850417749640057104620611, 6.75660847318802293730167157593, 7.892471043621819615920173398970, 8.566300400663981275163433547514, 9.955349734856393923715109351594, 10.85918309475042520460963692597, 11.59552154396932964427208799062, 13.34084350441620912736220911853, 14.18285947674364106336908034027
