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 | \( 1 + (-113 + 113i)T - 9.40e3iT^{2} \) |
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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
−14.07678509670913698784115097051, −13.17493378571447988228762422563, −11.43258225213499663048712549907, −10.72502973846190124713599141829, −9.597015855697787738483434200141, −8.367731524258278712629340586843, −7.51340445053640932574810495743, −6.18924078400127978636880539657, −3.98132795514445681081630975342, −3.24997244599020296648130090940,
1.00988996578530781446821528905, 2.55924115136985596871101929537, 5.06184352957247083015334092420, 6.18178216206250038301597745392, 7.82499744274678226631490166019, 8.908446731241995236782187308874, 9.543373303150697688468989652902, 11.14164475424363069891522428439, 12.31858788737258624574846345451, 12.76639049567333604254548808238