| L(s) = 1 | + (0.415 − 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (3.08 − 1.98i)5-s + (0.654 − 0.755i)6-s + (0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.522 − 3.63i)10-s + (2.02 + 4.42i)11-s + (−0.415 − 0.909i)12-s + (0.731 + 5.08i)13-s + (−0.841 − 0.540i)14-s + (3.52 − 1.03i)15-s + (−0.142 + 0.989i)16-s + (3.78 − 4.36i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (1.38 − 0.887i)5-s + (0.267 − 0.308i)6-s + (0.0537 − 0.374i)7-s + (−0.339 + 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.165 − 1.14i)10-s + (0.609 + 1.33i)11-s + (−0.119 − 0.262i)12-s + (0.202 + 1.41i)13-s + (−0.224 − 0.144i)14-s + (0.909 − 0.267i)15-s + (−0.0355 + 0.247i)16-s + (0.917 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.46339 - 1.44630i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.46339 - 1.44630i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 23 | \( 1 + (0.945 + 4.70i)T \) |
| good | 5 | \( 1 + (-3.08 + 1.98i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (-2.02 - 4.42i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.731 - 5.08i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.78 + 4.36i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.81 - 2.09i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-0.00679 + 0.00784i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (8.51 - 2.50i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (2.25 + 1.44i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (4.92 - 3.16i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (8.75 + 2.56i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 + (-0.591 + 4.11i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.390 + 2.71i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (5.16 - 1.51i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-5.27 + 11.5i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (6.22 - 13.6i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (6.26 + 7.23i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.507 + 3.53i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-1.65 - 1.06i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-6.50 - 1.91i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-13.8 + 8.89i)T + (40.2 - 88.2i)T^{2} \) |
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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
−9.740349586303970118111481532984, −9.351716947935499460062987611417, −8.622522506145710213488170005131, −7.27307087385488852144199385349, −6.40882600806850842604919358262, −5.14770834691879768237968587458, −4.63042674418976094511883448003, −3.53115977005435663702901200214, −2.02812274519735395599875094493, −1.50991770497904203191490721792,
1.59440744832873086752446866315, 3.07638975758667075251229856612, 3.49717903725793876977709680928, 5.49172933529313028599452382812, 5.76613749041915730824362175419, 6.60818133178476105075472312917, 7.62395048990640131294626654819, 8.433923440637857180283053704271, 9.228774174986775227553185497627, 10.04898725007510126242355209266
