| L(s) = 1 | + (−1.99 + 0.00346i)2-s + (−1.67 − 0.448i)3-s + (3.99 − 0.0138i)4-s + (0.893 − 0.239i)5-s + (3.34 + 0.890i)6-s + (−6.85 − 1.43i)7-s + (−7.99 + 0.0415i)8-s + (2.59 + 1.50i)9-s + (−1.78 + 0.481i)10-s + (13.1 + 3.52i)11-s + (−6.69 − 1.76i)12-s + (−5.95 + 5.95i)13-s + (13.7 + 2.85i)14-s − 1.60·15-s + (15.9 − 0.110i)16-s + (−7.06 + 4.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.00173i)2-s + (−0.557 − 0.149i)3-s + (0.999 − 0.00346i)4-s + (0.178 − 0.0478i)5-s + (0.557 + 0.148i)6-s + (−0.978 − 0.205i)7-s + (−0.999 + 0.00519i)8-s + (0.288 + 0.166i)9-s + (−0.178 + 0.0481i)10-s + (1.19 + 0.320i)11-s + (−0.558 − 0.147i)12-s + (−0.457 + 0.457i)13-s + (0.978 + 0.203i)14-s − 0.106·15-s + (0.999 − 0.00692i)16-s + (−0.415 + 0.240i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.757814 + 0.125238i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.757814 + 0.125238i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 - 0.00346i)T \) |
| 3 | \( 1 + (1.67 + 0.448i)T \) |
| 7 | \( 1 + (6.85 + 1.43i)T \) |
| good | 5 | \( 1 + (-0.893 + 0.239i)T + (21.6 - 12.5i)T^{2} \) |
| 11 | \( 1 + (-13.1 - 3.52i)T + (104. + 60.5i)T^{2} \) |
| 13 | \( 1 + (5.95 - 5.95i)T - 169iT^{2} \) |
| 17 | \( 1 + (7.06 - 4.08i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (6.93 + 25.8i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (-17.5 - 10.1i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-23.5 - 23.5i)T + 841iT^{2} \) |
| 31 | \( 1 + (-22.8 + 13.1i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-10.0 - 37.6i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 69.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-49.4 + 49.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-64.1 - 37.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-36.8 - 9.88i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (5.08 - 18.9i)T + (-3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-1.81 - 6.76i)T + (-3.22e3 + 1.86e3i)T^{2} \) |
| 67 | \( 1 + (24.2 - 90.3i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 96.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-13.4 - 23.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (5.05 - 8.75i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (64.5 - 64.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-8.78 + 15.2i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 161. iT - 9.40e3T^{2} \) |
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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
−11.28125052596993158014795306040, −10.38245544216932257520850560064, −9.380996913670258646726051792577, −8.971104876138149861062373849641, −7.31936818559109871677887070449, −6.77176279023228478734953381483, −5.91357318899566484625062828849, −4.24517839746289872355737136927, −2.59029547046696108644693165879, −0.961132561965930869426456121806,
0.71777491095611300722475291005, 2.54378187648615403820451218420, 4.00815111078498674741509326986, 5.91029772345638037229182665944, 6.35890398198608195289435724293, 7.46787052166569152202395720339, 8.691675478429371818131057467117, 9.546202573329781167430543447629, 10.19580533936826579258477793330, 11.10904748404097837255791120137
