| L(s) = 1 | + (−1.23 − 0.680i)2-s + (−0.755 + 0.654i)3-s + (1.07 + 1.68i)4-s + (−3.34 + 0.480i)5-s + (1.38 − 0.297i)6-s + (−0.0697 + 0.152i)7-s + (−0.181 − 2.82i)8-s + (0.142 − 0.989i)9-s + (4.46 + 1.67i)10-s + (−0.816 + 2.77i)11-s + (−1.91 − 0.572i)12-s + (−0.257 + 0.117i)13-s + (0.190 − 0.141i)14-s + (2.21 − 2.55i)15-s + (−1.69 + 3.62i)16-s + (−0.333 − 0.214i)17-s + ⋯ |
| L(s) = 1 | + (−0.876 − 0.481i)2-s + (−0.436 + 0.378i)3-s + (0.536 + 0.843i)4-s + (−1.49 + 0.214i)5-s + (0.564 − 0.121i)6-s + (−0.0263 + 0.0577i)7-s + (−0.0643 − 0.997i)8-s + (0.0474 − 0.329i)9-s + (1.41 + 0.530i)10-s + (−0.246 + 0.837i)11-s + (−0.553 − 0.165i)12-s + (−0.0715 + 0.0326i)13-s + (0.0509 − 0.0379i)14-s + (0.570 − 0.658i)15-s + (−0.423 + 0.905i)16-s + (−0.0809 − 0.0519i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0244 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0244 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.244326 - 0.238429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.244326 - 0.238429i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 0.680i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (3.99 + 2.66i)T \) |
| good | 5 | \( 1 + (3.34 - 0.480i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.0697 - 0.152i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.816 - 2.77i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.257 - 0.117i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.333 + 0.214i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.495 - 0.770i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-4.21 + 6.56i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-7.04 + 8.13i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (8.61 + 1.23i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.340 + 2.36i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-7.64 + 6.62i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 + (4.46 + 2.04i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-5.23 + 2.39i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (4.46 + 3.87i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (3.11 + 10.6i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (4.45 - 1.30i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (11.0 - 7.09i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.02 + 2.25i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-9.74 - 1.40i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (1.33 + 1.54i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.942 + 6.55i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.51534770478856515338803110133, −9.919193169302150089259946383822, −8.814307817058745306866802705053, −7.892562734788128016700106324099, −7.32201625587849995810110692937, −6.21610028310261390772386478863, −4.49166273584066152523841050529, −3.83641816656183107201078491479, −2.45943805911349986364479647714, −0.33869530422333529712633678997,
1.07594090316076189792048583204, 3.08216586014877355293768649915, 4.58574572849714845075838564483, 5.65746868987211949387344742869, 6.76247488371442922004261649480, 7.49905918067407452322052060211, 8.305535893592592704665198146225, 8.857687943684298052394669937079, 10.30426455757858966052878756821, 10.88757263589935546924716812247
