| L(s) = 1 | + (−0.468 + 2.47i)2-s + (−0.114 + 3.06i)3-s + (−4.06 − 1.59i)4-s + (1.36 + 1.77i)5-s + (−7.54 − 1.72i)6-s + (1.10 − 2.40i)7-s + (3.17 − 5.04i)8-s + (−6.39 − 0.479i)9-s + (−5.03 + 2.54i)10-s + (0.241 + 3.22i)11-s + (5.35 − 12.2i)12-s + (1.81 − 0.636i)13-s + (5.44 + 3.86i)14-s + (−5.58 + 3.97i)15-s + (4.62 + 4.28i)16-s + (1.00 + 1.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.331 + 1.75i)2-s + (−0.0662 + 1.77i)3-s + (−2.03 − 0.796i)4-s + (0.609 + 0.792i)5-s + (−3.08 − 0.703i)6-s + (0.417 − 0.908i)7-s + (1.12 − 1.78i)8-s + (−2.13 − 0.159i)9-s + (−1.59 + 0.806i)10-s + (0.0729 + 0.972i)11-s + (1.54 − 3.54i)12-s + (0.504 − 0.176i)13-s + (1.45 + 1.03i)14-s + (−1.44 + 1.02i)15-s + (1.15 + 1.07i)16-s + (0.242 + 0.328i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.524067 - 0.853751i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.524067 - 0.853751i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.36 - 1.77i)T \) |
| 7 | \( 1 + (-1.10 + 2.40i)T \) |
| good | 2 | \( 1 + (0.468 - 2.47i)T + (-1.86 - 0.730i)T^{2} \) |
| 3 | \( 1 + (0.114 - 3.06i)T + (-2.99 - 0.224i)T^{2} \) |
| 11 | \( 1 + (-0.241 - 3.22i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (-1.81 + 0.636i)T + (10.1 - 8.10i)T^{2} \) |
| 17 | \( 1 + (-1.00 - 1.35i)T + (-5.01 + 16.2i)T^{2} \) |
| 19 | \( 1 + (-2.52 + 4.36i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.11 - 3.77i)T + (6.77 + 21.9i)T^{2} \) |
| 29 | \( 1 + (2.61 + 2.08i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (1.27 - 0.734i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.48 + 2.39i)T + (25.1 + 27.1i)T^{2} \) |
| 41 | \( 1 + (-1.44 + 0.329i)T + (36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-6.56 + 4.12i)T + (18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (8.59 + 1.62i)T + (43.7 + 17.1i)T^{2} \) |
| 53 | \( 1 + (0.904 - 0.394i)T + (36.0 - 38.8i)T^{2} \) |
| 59 | \( 1 + (1.70 - 0.524i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (7.28 - 2.86i)T + (44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-12.0 - 3.21i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (3.77 + 4.72i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-2.81 + 0.533i)T + (67.9 - 26.6i)T^{2} \) |
| 79 | \( 1 + (-11.5 - 6.66i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.95 + 11.3i)T + (-64.8 - 51.7i)T^{2} \) |
| 89 | \( 1 + (0.579 - 7.73i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (-6.20 - 6.20i)T + 97iT^{2} \) |
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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
−13.39135754721504060418394271803, −11.19844726819580717255015412079, −10.45962106406394696162525239594, −9.615478840467700279429135181141, −9.045142532739405841407726320500, −7.66626525456848301955162328309, −6.79019527779350579297703891269, −5.53207672104969271093693988129, −4.80575310967878707834032572164, −3.63228921043422691280540604616,
1.01910904903769902975370708584, 1.90939749743392529029521837331, 3.09854992234970115121818780971, 5.18291375555772726751080623389, 6.22265036424588359416693918081, 8.009928195717613747230273006494, 8.654709429238526540503035940342, 9.394963823405276413376701758593, 10.92782959171396353091138619330, 11.63687605640806714746835410221
