| L(s) = 1 | + (−1.30 − 0.887i)3-s + (−3.55 − 0.266i)5-s + (−2.60 − 0.454i)7-s + (−0.189 − 0.483i)9-s + (−3.28 − 1.29i)11-s + (1.33 − 1.06i)13-s + (4.38 + 3.49i)15-s + (4.60 + 4.95i)17-s + (−1.26 + 2.18i)19-s + (2.98 + 2.90i)21-s + (4.90 − 5.29i)23-s + (7.59 + 1.14i)25-s + (−1.23 + 5.40i)27-s + (0.815 + 3.57i)29-s + (−2.53 − 4.39i)31-s + ⋯ |
| L(s) = 1 | + (−0.751 − 0.512i)3-s + (−1.58 − 0.119i)5-s + (−0.985 − 0.171i)7-s + (−0.0632 − 0.161i)9-s + (−0.991 − 0.389i)11-s + (0.370 − 0.295i)13-s + (1.13 + 0.903i)15-s + (1.11 + 1.20i)17-s + (−0.289 + 0.501i)19-s + (0.652 + 0.633i)21-s + (1.02 − 1.10i)23-s + (1.51 + 0.229i)25-s + (−0.237 + 1.04i)27-s + (0.151 + 0.663i)29-s + (−0.455 − 0.789i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.330471 + 0.131157i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.330471 + 0.131157i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.60 + 0.454i)T \) |
| good | 3 | \( 1 + (1.30 + 0.887i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (3.55 + 0.266i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (3.28 + 1.29i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-1.33 + 1.06i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-4.60 - 4.95i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (1.26 - 2.18i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.90 + 5.29i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.815 - 3.57i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (2.53 + 4.39i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.30 + 0.712i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-3.85 - 8.01i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-0.185 + 0.386i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (12.8 - 1.93i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (3.74 - 1.15i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.392 - 5.24i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-3.85 + 12.4i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-8.82 + 5.09i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.4 - 2.60i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.665 + 4.41i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-5.38 - 3.10i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.83 - 7.31i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (7.37 - 2.89i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 5.29iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.79113234707731464666305699121, −9.622066938773278813819432656501, −8.321842196034042863138123201828, −7.926841023779227847546759922162, −6.84525521489608004886338703136, −6.14757921307521766065716055336, −5.12887689112767910594544651178, −3.78145162481093639620915386880, −3.15191625198596553023992608736, −0.836517559549957832420847826639,
0.28965140721090699471056902015, 2.86854683954675614163508273964, 3.72824241675231609741077614990, 4.86929224173547063894421863439, 5.50886524308176744683026813712, 6.88575671021804785835325617797, 7.49152666811474138408407294258, 8.411189666786200732432873061184, 9.518317705768764440064750762270, 10.28591280031696535354273477980
