L(s) = 1 | + (4.46 + 2.57i)5-s + (5.25 − 3.03i)11-s + 12.5·13-s + (−14.9 + 8.64i)17-s + (−12.9 + 22.4i)19-s + (2.09 + 1.20i)23-s + (0.791 + 1.37i)25-s + 55.8i·29-s + (7.64 + 13.2i)31-s + (11.8 − 20.5i)37-s − 15.4i·41-s + 27.7·43-s + (24.6 + 14.2i)47-s + (−40.4 + 23.3i)53-s + 31.2·55-s + ⋯ |
L(s) = 1 | + (0.893 + 0.515i)5-s + (0.477 − 0.275i)11-s + 0.967·13-s + (−0.881 + 0.508i)17-s + (−0.680 + 1.17i)19-s + (0.0909 + 0.0525i)23-s + (0.0316 + 0.0548i)25-s + 1.92i·29-s + (0.246 + 0.427i)31-s + (0.320 − 0.555i)37-s − 0.377i·41-s + 0.645·43-s + (0.524 + 0.302i)47-s + (−0.762 + 0.440i)53-s + 0.568·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.257273005\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.257273005\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-4.46 - 2.57i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-5.25 + 3.03i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 12.5T + 169T^{2} \) |
| 17 | \( 1 + (14.9 - 8.64i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (12.9 - 22.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-2.09 - 1.20i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 55.8iT - 841T^{2} \) |
| 31 | \( 1 + (-7.64 - 13.2i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-11.8 + 20.5i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 15.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 27.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-24.6 - 14.2i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (40.4 - 23.3i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-69.8 + 40.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (31.3 - 54.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (58.0 + 100. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 49.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-15.3 - 26.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (71.3 - 123. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 28.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-145. - 84.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 113.T + 9.40e3T^{2} \) |
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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
−9.114858211054778359995492091063, −8.723047210943241563973635265950, −7.72750550346284972989440960489, −6.59008280600817937099802735763, −6.24798710392253447845777025688, −5.42130903012395102826610328481, −4.19239527384208418864425263828, −3.38558412352894232207873786085, −2.20077333846730210329843650925, −1.29854636209180201016643701182,
0.60113403828317595424369085794, 1.81268721949698941319463305847, 2.71038960915827705592273429316, 4.12698278106841830841458365190, 4.73453625580746821638546824853, 5.86802507742361212328942472118, 6.37041110830521199884769492333, 7.27468037456177108944856750515, 8.369172993604087828033024893121, 9.016179150476445515409911810375