L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 + 0.707i)8-s + (−3 − 1.73i)11-s + (3.34 + 0.896i)13-s + (0.500 − 0.866i)16-s + (−4.24 + 4.24i)17-s + 5i·19-s + (−0.896 + 3.34i)22-s + (−1.55 + 5.79i)23-s − 3.46i·26-s + (3.46 − 6i)29-s + (−2 − 3.46i)31-s + (−0.965 − 0.258i)32-s + (5.19 + 3i)34-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.249 + 0.249i)8-s + (−0.904 − 0.522i)11-s + (0.928 + 0.248i)13-s + (0.125 − 0.216i)16-s + (−1.02 + 1.02i)17-s + 1.14i·19-s + (−0.191 + 0.713i)22-s + (−0.323 + 1.20i)23-s − 0.679i·26-s + (0.643 − 1.11i)29-s + (−0.359 − 0.622i)31-s + (−0.170 − 0.0457i)32-s + (0.891 + 0.514i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8788164490\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8788164490\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.34 - 0.896i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (4.24 - 4.24i)T - 17iT^{2} \) |
| 19 | \( 1 - 5iT - 19T^{2} \) |
| 23 | \( 1 + (1.55 - 5.79i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-3.46 + 6i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.89 + 4.89i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.5 + 0.866i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.13 - 11.7i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-1.55 - 5.79i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.866 + 1.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.24 - 8.36i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 6.92iT - 71T^{2} \) |
| 73 | \( 1 + (8.57 - 8.57i)T - 73iT^{2} \) |
| 79 | \( 1 + (-12.1 - 7i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.69 + 2.32i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + (-5.01 + 1.34i)T + (84.0 - 48.5i)T^{2} \) |
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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
−9.818351686902663873461742586219, −8.975240209486102795401749395844, −8.189068438101343712570800140840, −7.63556085194914717132262358059, −6.21019722484517981741393524887, −5.69227125504992882350803230249, −4.33801283632997244411210065913, −3.66075578043579037988752687398, −2.49816346797713282965529281664, −1.37841394837615382744794933492,
0.39549994904944847830084005680, 2.17184774204511474164545603884, 3.37102128508124179314405231337, 4.76896603818387863779437275632, 5.11198249127094848023591878290, 6.44538714063074558911738872155, 6.92517134787697015368755049070, 7.83011921011230002570394578401, 8.771274967317130688248608368855, 9.111131546146960896808532892390