| L(s) = 1 | + (−1.93 + 0.503i)2-s + (−2.40 + 1.78i)3-s + (3.49 − 1.94i)4-s + (3.67 − 1.33i)5-s + (3.76 − 4.67i)6-s + (2.42 − 0.428i)7-s + (−5.78 + 5.52i)8-s + (2.61 − 8.61i)9-s + (−6.43 + 4.43i)10-s + (−6.11 + 16.8i)11-s + (−4.94 + 10.9i)12-s + (15.7 + 13.2i)13-s + (−4.48 + 2.05i)14-s + (−6.45 + 9.77i)15-s + (8.41 − 13.6i)16-s + (−11.4 + 19.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.967 + 0.251i)2-s + (−0.803 + 0.595i)3-s + (0.873 − 0.486i)4-s + (0.734 − 0.267i)5-s + (0.627 − 0.778i)6-s + (0.346 − 0.0611i)7-s + (−0.722 + 0.690i)8-s + (0.290 − 0.956i)9-s + (−0.643 + 0.443i)10-s + (−0.556 + 1.52i)11-s + (−0.411 + 0.911i)12-s + (1.21 + 1.01i)13-s + (−0.320 + 0.146i)14-s + (−0.430 + 0.651i)15-s + (0.525 − 0.850i)16-s + (−0.674 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.577742 + 0.503389i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.577742 + 0.503389i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.93 - 0.503i)T \) |
| 3 | \( 1 + (2.40 - 1.78i)T \) |
| good | 5 | \( 1 + (-3.67 + 1.33i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-2.42 + 0.428i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (6.11 - 16.8i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-15.7 - 13.2i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (11.4 - 19.8i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-6.65 + 3.83i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-1.94 - 0.343i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-33.6 + 28.2i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-20.3 - 3.58i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (15.3 - 26.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (4.57 + 3.83i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-10.9 + 29.9i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-53.8 + 9.49i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 19.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (19.2 + 52.7i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (2.78 + 15.7i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-1.41 + 1.68i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (86.0 + 49.7i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-31.8 - 55.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (12.9 + 15.4i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-33.4 - 39.9i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (8.30 + 14.3i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (120. + 43.6i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−13.78222256227571339440795537067, −12.35140421053007704031212714011, −11.29093440545617121324879119260, −10.33298292380434378701725187645, −9.582844872083683865282007966830, −8.496148623897395346510325255316, −6.88039924019176246977489539206, −5.91679534012101571686872939584, −4.53903143202361162614373939308, −1.71028220366578803628341691745,
0.915604898696525971167532016982, 2.80260955626902876804212398001, 5.57560123907036480986448297562, 6.45140772438189714640005240283, 7.82617986370912252161827181678, 8.786530495086881120973551962956, 10.38298323293895917881248724628, 10.93397605332657789488564297742, 11.84902870202889912577404392682, 13.17642851664977058774876905873
