| L(s) = 1 | + (−1.79 + 0.270i)2-s + (−0.0106 − 0.00723i)3-s + (1.23 − 0.382i)4-s + (0.0531 − 0.709i)5-s + (0.0210 + 0.0101i)6-s + (−2.62 + 0.358i)7-s + (1.15 − 0.554i)8-s + (−1.09 − 2.79i)9-s + (0.0965 + 1.28i)10-s + (−1.14 + 2.90i)11-s + (−0.0159 − 0.00490i)12-s + (0.623 + 0.781i)13-s + (4.60 − 1.35i)14-s + (−0.00569 + 0.00714i)15-s + (−4.05 + 2.76i)16-s + (−0.265 + 0.246i)17-s + ⋯ |
| L(s) = 1 | + (−1.26 + 0.191i)2-s + (−0.00612 − 0.00417i)3-s + (0.619 − 0.191i)4-s + (0.0237 − 0.317i)5-s + (0.00858 + 0.00413i)6-s + (−0.990 + 0.135i)7-s + (0.406 − 0.195i)8-s + (−0.365 − 0.930i)9-s + (0.0305 + 0.407i)10-s + (−0.343 + 0.876i)11-s + (−0.00459 − 0.00141i)12-s + (0.172 + 0.216i)13-s + (1.23 − 0.361i)14-s + (−0.00147 + 0.00184i)15-s + (−1.01 + 0.691i)16-s + (−0.0644 + 0.0597i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.378610 + 0.287192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378610 + 0.287192i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.62 - 0.358i)T \) |
| 13 | \( 1 + (-0.623 - 0.781i)T \) |
| good | 2 | \( 1 + (1.79 - 0.270i)T + (1.91 - 0.589i)T^{2} \) |
| 3 | \( 1 + (0.0106 + 0.00723i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.0531 + 0.709i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (1.14 - 2.90i)T + (-8.06 - 7.48i)T^{2} \) |
| 17 | \( 1 + (0.265 - 0.246i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.13 + 1.96i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.43 - 1.33i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.770 - 3.37i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.15 - 3.74i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.41 + 1.36i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (6.69 - 3.22i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-6.11 - 2.94i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-4.36 + 0.658i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (1.43 - 0.442i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-1.07 - 14.3i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-9.39 - 2.89i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-5.13 - 8.90i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.51 - 11.0i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-3.16 - 0.477i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (2.61 - 4.53i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (10.3 - 13.0i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (0.759 + 1.93i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 - 4.00T + 97T^{2} \) |
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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
−10.34271990497908760808375420136, −9.739842920054089378568756589784, −8.973807428470735338989685342241, −8.528259582849073919824112476826, −7.10267503457684297589271790758, −6.80702638804739419791753731160, −5.46246602830079042732694311684, −4.15988519132028291703282647026, −2.83074360195137078087445707419, −1.08572190983978924343543461032,
0.47962113454048670420308928091, 2.29202294177056848931876144828, 3.37631235098046028746877250090, 4.96238293206036239735582969215, 6.07470661090287971546452502371, 7.11523608028603868388850272396, 8.034657247344663440374529876668, 8.662283289222416346189397395650, 9.576052890007158589176144455920, 10.44290236354855010746066570943
