| L(s) = 1 | + (−0.996 − 0.0871i)2-s + (0.984 + 0.173i)4-s + (1.39 − 1.74i)5-s + (0.564 + 0.395i)7-s + (−0.965 − 0.258i)8-s + (−1.54 + 1.61i)10-s + (−1.76 + 4.83i)11-s + (0.446 + 5.10i)13-s + (−0.527 − 0.442i)14-s + (0.939 + 0.342i)16-s + (−0.221 + 0.0592i)17-s + (−6.21 + 3.58i)19-s + (1.67 − 1.47i)20-s + (2.17 − 4.66i)22-s + (4.94 + 7.06i)23-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.0616i)2-s + (0.492 + 0.0868i)4-s + (0.624 − 0.780i)5-s + (0.213 + 0.149i)7-s + (−0.341 − 0.0915i)8-s + (−0.488 + 0.511i)10-s + (−0.531 + 1.45i)11-s + (0.123 + 1.41i)13-s + (−0.141 − 0.118i)14-s + (0.234 + 0.0855i)16-s + (−0.0536 + 0.0143i)17-s + (−1.42 + 0.823i)19-s + (0.375 − 0.330i)20-s + (0.464 − 0.995i)22-s + (1.03 + 1.47i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.952334 + 0.521997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.952334 + 0.521997i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 0.0871i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.39 + 1.74i)T \) |
| good | 7 | \( 1 + (-0.564 - 0.395i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (1.76 - 4.83i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.446 - 5.10i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (0.221 - 0.0592i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (6.21 - 3.58i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.94 - 7.06i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-3.94 + 3.30i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.210 + 1.19i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.11 - 7.90i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.46 + 6.51i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.185 + 0.397i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-2.03 + 2.91i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (8.07 - 8.07i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.64 - 0.962i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.117 + 0.665i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.55 + 0.835i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.366 - 0.211i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.47 - 5.50i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.26 - 3.88i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.605 - 6.92i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (0.102 + 0.178i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.2 + 5.71i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.09170974468178579001840843914, −9.554356359190770837006217647669, −8.820670008201225908369726038647, −7.992650434061260206852887185836, −7.01686536805192372378533047989, −6.14837918725343245813506071366, −4.99696161115449746636670960840, −4.16413963383336029308242721969, −2.29761701449189385504790383203, −1.55633467612233912443148890156,
0.68699906011085838427002209029, 2.51114738770759514297464221116, 3.17919039367142211160950121577, 4.90945638381662024958660659988, 6.02166343700429314860955733121, 6.56321541695021596864996950882, 7.70556794766313892240039510796, 8.449497435110813501041251123651, 9.170911826031312961776615847232, 10.38148004413543483393735433453
