L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.29 − 2.70i)3-s − 3i·4-s + (−0.707 − 4.94i)5-s + (−2.82 + i)6-s + 7i·7-s + (−4.94 + 4.94i)8-s + (−5.65 − 7i)9-s + (−3.00 + 4.00i)10-s + 9.89i·11-s + (−8.12 − 3.87i)12-s + (8 − 8i)13-s + (4.94 − 4.94i)14-s + (−14.3 − 4.48i)15-s − 4.99·16-s + (18.3 − 18.3i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.353i)2-s + (0.430 − 0.902i)3-s − 0.750i·4-s + (−0.141 − 0.989i)5-s + (−0.471 + 0.166i)6-s + i·7-s + (−0.618 + 0.618i)8-s + (−0.628 − 0.777i)9-s + (−0.300 + 0.400i)10-s + 0.899i·11-s + (−0.676 − 0.323i)12-s + (0.615 − 0.615i)13-s + (0.353 − 0.353i)14-s + (−0.954 − 0.299i)15-s − 0.312·16-s + (1.08 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.406414 - 1.09037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.406414 - 1.09037i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.29 + 2.70i)T \) |
| 5 | \( 1 + (0.707 + 4.94i)T \) |
| 7 | \( 1 - 7iT \) |
good | 2 | \( 1 + (0.707 + 0.707i)T + 4iT^{2} \) |
| 11 | \( 1 - 9.89iT - 121T^{2} \) |
| 13 | \( 1 + (-8 + 8i)T - 169iT^{2} \) |
| 17 | \( 1 + (-18.3 + 18.3i)T - 289iT^{2} \) |
| 19 | \( 1 + 10T + 361T^{2} \) |
| 23 | \( 1 + (-24.0 + 24.0i)T - 529iT^{2} \) |
| 29 | \( 1 - 4.24T + 841T^{2} \) |
| 31 | \( 1 - 14iT - 961T^{2} \) |
| 37 | \( 1 + (-30 + 30i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 33.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36 - 36i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-4.24 + 4.24i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (70.7 - 70.7i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 29.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 14iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-32 + 32i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 59.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-39 + 39i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 56iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-36.7 - 36.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 19.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-113 - 113i)T + 9.40e3iT^{2} \) |
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
−12.76639049567333604254548808238, −12.31858788737258624574846345451, −11.14164475424363069891522428439, −9.543373303150697688468989652902, −8.908446731241995236782187308874, −7.82499744274678226631490166019, −6.18178216206250038301597745392, −5.06184352957247083015334092420, −2.55924115136985596871101929537, −1.00988996578530781446821528905,
3.24997244599020296648130090940, 3.98132795514445681081630975342, 6.18924078400127978636880539657, 7.51340445053640932574810495743, 8.367731524258278712629340586843, 9.597015855697787738483434200141, 10.72502973846190124713599141829, 11.43258225213499663048712549907, 13.17493378571447988228762422563, 14.07678509670913698784115097051