| L(s) = 1 | + (−3.30 − 0.504i)3-s + (−2.26 − 2.68i)5-s + (1.34 − 1.87i)7-s + (7.83 + 2.44i)9-s + (4.12 − 3.89i)11-s + (0.147 − 1.10i)13-s + (6.14 + 10.0i)15-s + (2.68 − 4.03i)17-s + (−7.00 − 1.61i)19-s + (−5.40 + 5.50i)21-s + (−0.971 − 4.60i)23-s + (−1.24 + 7.25i)25-s + (−15.6 − 7.64i)27-s + (−1.17 + 5.57i)29-s + (−0.811 + 0.155i)31-s + ⋯ |
| L(s) = 1 | + (−1.91 − 0.291i)3-s + (−1.01 − 1.20i)5-s + (0.509 − 0.707i)7-s + (2.61 + 0.815i)9-s + (1.24 − 1.17i)11-s + (0.0408 − 0.306i)13-s + (1.58 + 2.59i)15-s + (0.650 − 0.978i)17-s + (−1.60 − 0.371i)19-s + (−1.17 + 1.20i)21-s + (−0.202 − 0.959i)23-s + (−0.249 + 1.45i)25-s + (−3.01 − 1.47i)27-s + (−0.218 + 1.03i)29-s + (−0.145 + 0.0279i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00162627 + 0.543707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00162627 + 0.543707i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (-3.89 + 12.3i)T \) |
| good | 3 | \( 1 + (3.30 + 0.504i)T + (2.86 + 0.894i)T^{2} \) |
| 5 | \( 1 + (2.26 + 2.68i)T + (-0.847 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.34 + 1.87i)T + (-2.21 - 6.64i)T^{2} \) |
| 11 | \( 1 + (-4.12 + 3.89i)T + (0.624 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.147 + 1.10i)T + (-12.5 - 3.40i)T^{2} \) |
| 17 | \( 1 + (-2.68 + 4.03i)T + (-6.57 - 15.6i)T^{2} \) |
| 19 | \( 1 + (7.00 + 1.61i)T + (17.0 + 8.33i)T^{2} \) |
| 23 | \( 1 + (0.971 + 4.60i)T + (-21.0 + 9.30i)T^{2} \) |
| 29 | \( 1 + (1.17 - 5.57i)T + (-26.5 - 11.7i)T^{2} \) |
| 31 | \( 1 + (0.811 - 0.155i)T + (28.8 - 11.4i)T^{2} \) |
| 37 | \( 1 + (-6.51 + 2.03i)T + (30.4 - 21.0i)T^{2} \) |
| 41 | \( 1 + (-6.16 + 2.45i)T + (29.8 - 28.1i)T^{2} \) |
| 43 | \( 1 + (5.99 + 2.12i)T + (33.4 + 27.0i)T^{2} \) |
| 47 | \( 1 + (0.401 - 1.59i)T + (-41.4 - 22.2i)T^{2} \) |
| 53 | \( 1 + (-2.67 - 2.00i)T + (14.8 + 50.8i)T^{2} \) |
| 59 | \( 1 + (-5.27 - 7.92i)T + (-22.8 + 54.4i)T^{2} \) |
| 61 | \( 1 + (3.18 + 4.07i)T + (-14.8 + 59.1i)T^{2} \) |
| 67 | \( 1 + (-0.337 + 0.400i)T + (-11.3 - 66.0i)T^{2} \) |
| 71 | \( 1 + (0.366 + 3.86i)T + (-69.7 + 13.3i)T^{2} \) |
| 73 | \( 1 + (-3.53 + 6.29i)T + (-38.0 - 62.2i)T^{2} \) |
| 79 | \( 1 + (9.00 + 0.341i)T + (78.7 + 5.97i)T^{2} \) |
| 83 | \( 1 + (-2.05 + 0.558i)T + (71.6 - 41.9i)T^{2} \) |
| 89 | \( 1 + (8.86 - 14.5i)T + (-40.5 - 79.2i)T^{2} \) |
| 97 | \( 1 + (15.2 + 2.92i)T + (90.1 + 35.8i)T^{2} \) |
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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
−10.55347425899904819086906078771, −9.170985007858155168191618879852, −8.220960940932043288261295021018, −7.29921744576974402451710148773, −6.42571825621142103492991588473, −5.48247083818994878618590412462, −4.55256123656040166770803964566, −4.02712950197201582565756546086, −1.15935671639784946820667634117, −0.46431015049952058397817989825,
1.75922194541197974628192698860, 3.94847565062311935053917530926, 4.35084091950416861646735049694, 5.73384661188524098688805842341, 6.44803023404512504059319848540, 7.09916147424420370258549407538, 8.118222475526360772407393394793, 9.654704030244919694671316438488, 10.27642725100520501393751141922, 11.28566315345670749581100725069
