| L(s) = 1 | + (0.793 + 0.111i)2-s + (−1.30 − 0.374i)4-s + (3.10 + 1.25i)5-s + (3.54 − 2.38i)7-s + (−2.45 − 1.09i)8-s + (2.32 + 1.34i)10-s + (−2.26 − 2.42i)11-s + (2.28 − 4.28i)13-s + (3.07 − 1.50i)14-s + (0.474 + 0.296i)16-s + (−5.16 − 1.09i)17-s + (0.636 − 1.42i)19-s + (−3.58 − 2.80i)20-s + (−1.52 − 2.17i)22-s + (1.01 + 2.79i)23-s + ⋯ |
| L(s) = 1 | + (0.561 + 0.0788i)2-s + (−0.652 − 0.187i)4-s + (1.38 + 0.561i)5-s + (1.33 − 0.903i)7-s + (−0.869 − 0.386i)8-s + (0.735 + 0.424i)10-s + (−0.682 − 0.730i)11-s + (0.632 − 1.18i)13-s + (0.822 − 0.401i)14-s + (0.118 + 0.0740i)16-s + (−1.25 − 0.266i)17-s + (0.145 − 0.327i)19-s + (−0.801 − 0.626i)20-s + (−0.325 − 0.463i)22-s + (0.211 + 0.581i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.19887 - 0.810000i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.19887 - 0.810000i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (2.26 + 2.42i)T \) |
| good | 2 | \( 1 + (-0.793 - 0.111i)T + (1.92 + 0.551i)T^{2} \) |
| 5 | \( 1 + (-3.10 - 1.25i)T + (3.59 + 3.47i)T^{2} \) |
| 7 | \( 1 + (-3.54 + 2.38i)T + (2.62 - 6.49i)T^{2} \) |
| 13 | \( 1 + (-2.28 + 4.28i)T + (-7.26 - 10.7i)T^{2} \) |
| 17 | \( 1 + (5.16 + 1.09i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.636 + 1.42i)T + (-12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.01 - 2.79i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.936 - 1.91i)T + (-17.8 - 22.8i)T^{2} \) |
| 31 | \( 1 + (-0.219 - 6.27i)T + (-30.9 + 2.16i)T^{2} \) |
| 37 | \( 1 + (-9.67 + 4.30i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (2.36 + 4.84i)T + (-25.2 + 32.3i)T^{2} \) |
| 43 | \( 1 + (3.85 + 4.58i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.60 - 5.59i)T + (-39.8 + 24.9i)T^{2} \) |
| 53 | \( 1 + (-4.45 - 1.44i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.20 + 0.797i)T + (52.0 - 27.6i)T^{2} \) |
| 61 | \( 1 + (2.32 + 0.0812i)T + (60.8 + 4.25i)T^{2} \) |
| 67 | \( 1 + (0.546 - 3.09i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.585 - 2.75i)T + (-64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-0.245 - 0.0258i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (1.58 - 11.2i)T + (-75.9 - 21.7i)T^{2} \) |
| 83 | \( 1 + (4.77 - 2.53i)T + (46.4 - 68.8i)T^{2} \) |
| 89 | \( 1 + (-13.5 + 7.79i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.52 - 11.1i)T + (-69.7 + 67.3i)T^{2} \) |
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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
−10.29485660000520953981938725221, −9.160079881033689095658020810597, −8.443888802874036758501675051330, −7.41979957173993192729130473695, −6.32131944372812139274026236467, −5.46407211442135937797542116324, −4.95699372913220194770192576254, −3.73294301022161510548262150152, −2.55571864662652789077741088490, −1.04430387281578829460349904831,
1.75678680649753093289989649735, 2.46404021302257208008082456014, 4.39347205397446696264061054765, 4.77479407341151900824543046593, 5.68482667643210388902300693149, 6.38002876687631085658506237425, 7.975360226723344386070356849942, 8.691786009309944623461438575773, 9.251412197584262801944032478270, 10.02217150685062744217440166115
