| L(s) = 1 | + (−1.57 − 0.726i)3-s + (−1.96 + 0.346i)5-s + (−0.910 + 1.57i)7-s + (1.94 + 2.28i)9-s + (4.10 − 2.37i)11-s + (0.151 + 0.415i)13-s + (3.34 + 0.883i)15-s + (−1.07 − 1.28i)17-s + (3.58 + 2.48i)19-s + (2.57 − 1.81i)21-s + (−5.93 − 1.04i)23-s + (−0.952 + 0.346i)25-s + (−1.39 − 5.00i)27-s + (−4.91 − 4.12i)29-s + (−4.88 − 2.82i)31-s + ⋯ |
| L(s) = 1 | + (−0.907 − 0.419i)3-s + (−0.879 + 0.155i)5-s + (−0.344 + 0.596i)7-s + (0.647 + 0.761i)9-s + (1.23 − 0.715i)11-s + (0.0419 + 0.115i)13-s + (0.863 + 0.228i)15-s + (−0.260 − 0.310i)17-s + (0.821 + 0.569i)19-s + (0.562 − 0.396i)21-s + (−1.23 − 0.218i)23-s + (−0.190 + 0.0693i)25-s + (−0.268 − 0.963i)27-s + (−0.913 − 0.766i)29-s + (−0.877 − 0.506i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.127445 - 0.346194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.127445 - 0.346194i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.57 + 0.726i)T \) |
| 19 | \( 1 + (-3.58 - 2.48i)T \) |
| good | 5 | \( 1 + (1.96 - 0.346i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.910 - 1.57i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.10 + 2.37i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.151 - 0.415i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.07 + 1.28i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (5.93 + 1.04i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.91 + 4.12i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (4.88 + 2.82i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.80iT - 37T^{2} \) |
| 41 | \( 1 + (-3.75 - 1.36i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (2.15 + 12.2i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (6.92 - 8.25i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.424 + 2.40i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (3.87 - 3.24i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.80 + 10.2i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.27 - 6.28i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.897 + 5.08i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (13.5 + 4.94i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-3.23 + 8.88i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.523 - 0.302i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.07 + 1.48i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.64 - 1.96i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−9.750997295476765129018566513611, −8.977689818596657488895039810221, −7.85525215363437323513895568908, −7.25108307427155197324547892446, −6.11952047689741339272330398042, −5.73841863216621305530675623793, −4.30758502696404370824515578395, −3.51473306585293972295409269495, −1.86096407377159421741911552170, −0.21233180215990438934002184689,
1.37122382989215664909076134197, 3.56273688142891468455228232186, 4.10831606524511559422981685922, 5.02050490571532852105067932227, 6.18935062368203110378942279293, 6.97656854183211876975329887631, 7.67213337601767471658889667036, 8.935972227792773549579487396922, 9.711172343852998208420546627166, 10.38878643320810374464025079802
