| L(s) = 1 | + (−2.53 + 0.102i)2-s + (−0.278 − 0.960i)3-s + (4.44 − 0.358i)4-s + (1.66 − 0.629i)5-s + (0.804 + 2.41i)6-s + (−2.05 + 4.81i)7-s + (−6.20 + 0.753i)8-s + (−0.845 + 0.534i)9-s + (−4.15 + 1.76i)10-s + (0.626 − 0.990i)11-s + (−1.58 − 4.17i)12-s + (−3.48 − 0.936i)13-s + (4.71 − 12.4i)14-s + (−1.06 − 1.41i)15-s + (6.88 − 1.11i)16-s + (−4.86 − 2.07i)17-s + ⋯ |
| L(s) = 1 | + (−1.79 + 0.0723i)2-s + (−0.160 − 0.554i)3-s + (2.22 − 0.179i)4-s + (0.742 − 0.281i)5-s + (0.328 + 0.984i)6-s + (−0.775 + 1.81i)7-s + (−2.19 + 0.266i)8-s + (−0.281 + 0.178i)9-s + (−1.31 + 0.559i)10-s + (0.188 − 0.298i)11-s + (−0.456 − 1.20i)12-s + (−0.965 − 0.259i)13-s + (1.26 − 3.32i)14-s + (−0.275 − 0.366i)15-s + (1.72 − 0.279i)16-s + (−1.17 − 0.502i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0100750 - 0.0895851i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0100750 - 0.0895851i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.278 + 0.960i)T \) |
| 13 | \( 1 + (3.48 + 0.936i)T \) |
| good | 2 | \( 1 + (2.53 - 0.102i)T + (1.99 - 0.160i)T^{2} \) |
| 5 | \( 1 + (-1.66 + 0.629i)T + (3.74 - 3.31i)T^{2} \) |
| 7 | \( 1 + (2.05 - 4.81i)T + (-4.84 - 5.04i)T^{2} \) |
| 11 | \( 1 + (-0.626 + 0.990i)T + (-4.71 - 9.93i)T^{2} \) |
| 17 | \( 1 + (4.86 + 2.07i)T + (11.7 + 12.2i)T^{2} \) |
| 19 | \( 1 + (-0.350 - 0.202i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.21 + 2.10i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.322 + 8.00i)T + (-28.9 + 2.33i)T^{2} \) |
| 31 | \( 1 + (-3.37 + 3.80i)T + (-3.73 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-1.51 - 0.309i)T + (34.0 + 14.5i)T^{2} \) |
| 41 | \( 1 + (9.64 - 2.79i)T + (34.6 - 21.9i)T^{2} \) |
| 43 | \( 1 + (1.80 + 8.85i)T + (-39.5 + 16.8i)T^{2} \) |
| 47 | \( 1 + (7.01 + 4.84i)T + (16.6 + 43.9i)T^{2} \) |
| 53 | \( 1 + (-0.532 - 4.38i)T + (-51.4 + 12.6i)T^{2} \) |
| 59 | \( 1 + (-0.739 + 4.55i)T + (-55.9 - 18.6i)T^{2} \) |
| 61 | \( 1 + (0.780 + 0.586i)T + (16.9 + 58.5i)T^{2} \) |
| 67 | \( 1 + (0.245 - 3.03i)T + (-66.1 - 10.7i)T^{2} \) |
| 71 | \( 1 + (-5.05 - 4.85i)T + (2.85 + 70.9i)T^{2} \) |
| 73 | \( 1 + (-1.42 - 2.70i)T + (-41.4 + 60.0i)T^{2} \) |
| 79 | \( 1 + (6.95 - 10.0i)T + (-28.0 - 73.8i)T^{2} \) |
| 83 | \( 1 + (-2.33 - 9.46i)T + (-73.4 + 38.5i)T^{2} \) |
| 89 | \( 1 + (14.6 - 8.47i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.9 + 8.95i)T + (19.4 - 95.0i)T^{2} \) |
| 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
−9.918967617153760463139319047622, −9.637635265914347333309175380558, −8.742110658070010289532269172913, −8.177437585847748389061945936749, −6.87089950692736547150485785667, −6.22896695708036411204452182947, −5.36138826140931430704521266324, −2.60617978875754033109245059941, −2.06380389028709035727884689809, −0.090127446630019628498889466138,
1.64889516535279752061395840204, 3.17106488792396038258064105018, 4.59135998501244970388062257708, 6.45889086247185437501696063931, 6.84815282806696069833857363247, 7.78209738519058929243319713124, 8.990476916995593976282602423201, 9.764853593313339812213868158000, 10.20758801881192464813523580074, 10.72734912915303876632123316347
