| L(s) = 1 | + (0.809 − 1.40i)2-s + (0.309 − 1.75i)3-s + (−0.310 − 0.538i)4-s + (−1.08 + 0.395i)5-s + (−2.21 − 1.85i)6-s + (−1.51 + 0.550i)7-s + 2.23·8-s + (−0.170 − 0.0621i)9-s + (−0.325 + 1.84i)10-s + (0.407 + 2.31i)11-s + (−1.04 + 0.379i)12-s + (−3.73 + 1.35i)13-s + (−0.452 + 2.56i)14-s + (0.358 + 2.03i)15-s + (2.42 − 4.20i)16-s + (1.10 − 1.90i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.991i)2-s + (0.178 − 1.01i)3-s + (−0.155 − 0.269i)4-s + (−0.486 + 0.176i)5-s + (−0.903 − 0.757i)6-s + (−0.571 + 0.208i)7-s + 0.788·8-s + (−0.0569 − 0.0207i)9-s + (−0.102 + 0.583i)10-s + (0.122 + 0.696i)11-s + (−0.300 + 0.109i)12-s + (−1.03 + 0.376i)13-s + (−0.120 + 0.685i)14-s + (0.0925 + 0.524i)15-s + (0.607 − 1.05i)16-s + (0.267 − 0.462i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.905933 - 1.00945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.905933 - 1.00945i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (8.72 - 5.73i)T \) |
| good | 2 | \( 1 + (-0.809 + 1.40i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.309 + 1.75i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (1.08 - 0.395i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.51 - 0.550i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.407 - 2.31i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (3.73 - 1.35i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.10 + 1.90i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.54 + 2.67i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.57 - 2.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.15 - 6.55i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.12 + 0.410i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (8.73 + 3.18i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + 5.68T + 41T^{2} \) |
| 43 | \( 1 + (-2.07 + 3.60i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.76 + 2.31i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-7.48 + 2.72i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.06 - 6.05i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (5.81 + 4.88i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (7.19 + 2.61i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.97 + 5.14i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.05 + 11.6i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-9.05 - 3.29i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.0946 + 0.536i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-11.0 - 9.25i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-16.0 + 5.82i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.17176942235054070182691004700, −12.22970252175785802568505183374, −11.88452182904195367431990524628, −10.50862565106247401111033948775, −9.303655767766768224524315440518, −7.46031906608914718675751850664, −7.08079802182788181515949444917, −4.94351292031495501648784080939, −3.36495736951116643092929115098, −1.98498828544889292198642657668,
3.59883906646058086193724671305, 4.67688129147468989758217996756, 5.89726389146520098915508164726, 7.20456253443029456682663193854, 8.354649223216683768515068929886, 9.836532417896936128521323352507, 10.52450533915312710273271390038, 12.04334999554147335687607496097, 13.23216432493516642741815626021, 14.31957727434009318274529268625
