| L(s) = 1 | + (0.301 + 1.15i)2-s + (−0.169 − 0.985i)3-s + (0.493 − 0.275i)4-s + (0.265 − 1.23i)5-s + (1.09 − 0.493i)6-s + (0.174 + 4.11i)7-s + (2.12 + 2.21i)8-s + (−0.942 + 0.333i)9-s + (1.51 − 0.0641i)10-s + (0.557 − 0.997i)11-s + (−0.354 − 0.439i)12-s + (−0.188 − 0.139i)13-s + (−4.71 + 1.44i)14-s + (−1.26 − 0.0535i)15-s + (−1.33 + 2.17i)16-s + (7.93 − 1.36i)17-s + ⋯ |
| L(s) = 1 | + (0.213 + 0.819i)2-s + (−0.0975 − 0.569i)3-s + (0.246 − 0.137i)4-s + (0.118 − 0.551i)5-s + (0.445 − 0.201i)6-s + (0.0660 + 1.55i)7-s + (0.751 + 0.784i)8-s + (−0.314 + 0.111i)9-s + (0.477 − 0.0202i)10-s + (0.168 − 0.300i)11-s + (−0.102 − 0.126i)12-s + (−0.0523 − 0.0387i)13-s + (−1.25 + 0.385i)14-s + (−0.325 − 0.0138i)15-s + (−0.334 + 0.543i)16-s + (1.92 − 0.329i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74219 + 0.601422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74219 + 0.601422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.169 + 0.985i)T \) |
| 149 | \( 1 + (-1.98 + 12.0i)T \) |
| good | 2 | \( 1 + (-0.301 - 1.15i)T + (-1.74 + 0.975i)T^{2} \) |
| 5 | \( 1 + (-0.265 + 1.23i)T + (-4.55 - 2.05i)T^{2} \) |
| 7 | \( 1 + (-0.174 - 4.11i)T + (-6.97 + 0.593i)T^{2} \) |
| 11 | \( 1 + (-0.557 + 0.997i)T + (-5.76 - 9.36i)T^{2} \) |
| 13 | \( 1 + (0.188 + 0.139i)T + (3.80 + 12.4i)T^{2} \) |
| 17 | \( 1 + (-7.93 + 1.36i)T + (16.0 - 5.66i)T^{2} \) |
| 19 | \( 1 + (2.51 + 1.70i)T + (7.08 + 17.6i)T^{2} \) |
| 23 | \( 1 + (-1.15 + 0.856i)T + (6.73 - 21.9i)T^{2} \) |
| 29 | \( 1 + (-4.14 - 4.71i)T + (-3.68 + 28.7i)T^{2} \) |
| 31 | \( 1 + (3.88 + 5.25i)T + (-9.07 + 29.6i)T^{2} \) |
| 37 | \( 1 + (4.08 + 2.28i)T + (19.3 + 31.5i)T^{2} \) |
| 41 | \( 1 + (4.82 + 2.97i)T + (18.4 + 36.6i)T^{2} \) |
| 43 | \( 1 + (10.2 - 1.31i)T + (41.6 - 10.8i)T^{2} \) |
| 47 | \( 1 + (-1.17 + 0.795i)T + (17.5 - 43.6i)T^{2} \) |
| 53 | \( 1 + (0.118 - 0.923i)T + (-51.2 - 13.3i)T^{2} \) |
| 59 | \( 1 + (4.31 - 1.73i)T + (42.5 - 40.8i)T^{2} \) |
| 61 | \( 1 + (2.08 - 0.543i)T + (53.2 - 29.7i)T^{2} \) |
| 67 | \( 1 + (1.13 - 3.71i)T + (-55.5 - 37.5i)T^{2} \) |
| 71 | \( 1 + (-1.05 - 0.227i)T + (64.6 + 29.2i)T^{2} \) |
| 73 | \( 1 + (8.04 + 0.684i)T + (71.9 + 12.3i)T^{2} \) |
| 79 | \( 1 + (0.301 + 3.54i)T + (-77.8 + 13.3i)T^{2} \) |
| 83 | \( 1 + (-0.453 + 5.32i)T + (-81.8 - 14.0i)T^{2} \) |
| 89 | \( 1 + (3.51 - 1.77i)T + (52.9 - 71.5i)T^{2} \) |
| 97 | \( 1 + (-14.0 - 1.79i)T + (93.8 + 24.4i)T^{2} \) |
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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
−11.46346521839435107446228148507, −10.30937827625861642427075845671, −9.010339386482393818085338033229, −8.401755837379099918009892460135, −7.41876188233210649564264103926, −6.41323646002827275494589961075, −5.53231657738350063492332914974, −5.07536344615613285585237400475, −2.94354591734737209128798717603, −1.60635103826153019039781346049,
1.39847665140700290968570478959, 3.15093543998876073993830169445, 3.80003864730654422938520492995, 4.87438089125396193078035428745, 6.46716022115083416693154973419, 7.25468408635332980565160815160, 8.183477141155320827795490994188, 9.867862792512602627158970831410, 10.30012374407076130336073143630, 10.77955205950699610856214013624
