| L(s) = 1 | + (2.17 − 0.286i)2-s + (−0.442 − 0.896i)3-s + (2.72 − 0.729i)4-s + (1.07 − 0.945i)5-s + (−1.21 − 1.82i)6-s + (0.241 − 2.63i)7-s + (1.65 − 0.686i)8-s + (−0.608 + 0.793i)9-s + (2.07 − 2.36i)10-s + (−4.77 + 0.312i)11-s + (−1.85 − 2.11i)12-s + (4.50 + 4.50i)13-s + (−0.230 − 5.80i)14-s + (−1.32 − 0.548i)15-s + (−1.47 + 0.849i)16-s + (4.08 + 0.569i)17-s + ⋯ |
| L(s) = 1 | + (1.53 − 0.202i)2-s + (−0.255 − 0.517i)3-s + (1.36 − 0.364i)4-s + (0.482 − 0.422i)5-s + (−0.497 − 0.745i)6-s + (0.0911 − 0.995i)7-s + (0.585 − 0.242i)8-s + (−0.202 + 0.264i)9-s + (0.656 − 0.748i)10-s + (−1.43 + 0.0943i)11-s + (−0.536 − 0.611i)12-s + (1.24 + 1.24i)13-s + (−0.0615 − 1.55i)14-s + (−0.342 − 0.141i)15-s + (−0.367 + 0.212i)16-s + (0.990 + 0.138i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.43327 - 1.43129i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43327 - 1.43129i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.442 + 0.896i)T \) |
| 7 | \( 1 + (-0.241 + 2.63i)T \) |
| 17 | \( 1 + (-4.08 - 0.569i)T \) |
| good | 2 | \( 1 + (-2.17 + 0.286i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (-1.07 + 0.945i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (4.77 - 0.312i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-4.50 - 4.50i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.367 + 2.79i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-1.14 - 0.563i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-0.234 - 1.17i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-1.66 + 0.819i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.312 + 4.76i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (2.29 - 11.5i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-2.19 - 5.28i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-1.67 + 6.26i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.55 + 2.02i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (10.5 + 1.39i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (3.94 + 11.6i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-6.44 - 3.71i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.14 + 3.43i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-1.15 - 0.391i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (7.56 - 15.3i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (2.46 - 5.95i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (8.90 + 2.38i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-6.87 + 1.36i)T + (89.6 - 37.1i)T^{2} \) |
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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
−11.34790405407015850302004896708, −10.93829619270856144459701492620, −9.661189167891264672637876721681, −8.282977904086480427007762141056, −7.14334033590226109017546914959, −6.19065737470571533828261687409, −5.27470030394272585284851769418, −4.39485677389778013858927772862, −3.14304744763352530651151024428, −1.59894377841986514548915099017,
2.64605956219805954421296011702, 3.42523423202066914092145817897, 4.90528816056389194764517914427, 5.79981010581113487744293266483, 6.00286584435868646686613379094, 7.67541218233332381768427426570, 8.750172638196464669529319330837, 10.19973055803799502408752638088, 10.73094170953463970693257187243, 11.91545481809954320780700069707
