L(s) = 1 | + (0.218 − 1.39i)2-s + (0.262 − 1.71i)3-s + (−1.90 − 0.611i)4-s + 0.130·5-s + (−2.33 − 0.740i)6-s − 3.77i·7-s + (−1.27 + 2.52i)8-s + (−2.86 − 0.897i)9-s + (0.0285 − 0.182i)10-s + 4.36i·11-s + (−1.54 + 3.10i)12-s − 5.50i·13-s + (−5.27 − 0.826i)14-s + (0.0342 − 0.223i)15-s + (3.25 + 2.32i)16-s − 0.402i·17-s + ⋯ |
L(s) = 1 | + (0.154 − 0.987i)2-s + (0.151 − 0.988i)3-s + (−0.952 − 0.305i)4-s + 0.0583·5-s + (−0.953 − 0.302i)6-s − 1.42i·7-s + (−0.449 + 0.893i)8-s + (−0.954 − 0.299i)9-s + (0.00902 − 0.0576i)10-s + 1.31i·11-s + (−0.446 + 0.894i)12-s − 1.52i·13-s + (−1.41 − 0.220i)14-s + (0.00883 − 0.0577i)15-s + (0.813 + 0.582i)16-s − 0.0975i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.344639 + 1.07988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.344639 + 1.07988i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.218 + 1.39i)T \) |
| 3 | \( 1 + (-0.262 + 1.71i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 0.130T + 5T^{2} \) |
| 7 | \( 1 + 3.77iT - 7T^{2} \) |
| 11 | \( 1 - 4.36iT - 11T^{2} \) |
| 13 | \( 1 + 5.50iT - 13T^{2} \) |
| 17 | \( 1 + 0.402iT - 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 29 | \( 1 - 8.77T + 29T^{2} \) |
| 31 | \( 1 - 7.75iT - 31T^{2} \) |
| 37 | \( 1 + 10.1iT - 37T^{2} \) |
| 41 | \( 1 + 2.80iT - 41T^{2} \) |
| 43 | \( 1 - 3.07T + 43T^{2} \) |
| 47 | \( 1 + 3.41T + 47T^{2} \) |
| 53 | \( 1 - 0.336T + 53T^{2} \) |
| 59 | \( 1 + 9.68iT - 59T^{2} \) |
| 61 | \( 1 + 2.78iT - 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 - 0.417T + 73T^{2} \) |
| 79 | \( 1 + 3.09iT - 79T^{2} \) |
| 83 | \( 1 + 3.59iT - 83T^{2} \) |
| 89 | \( 1 - 11.1iT - 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−10.39481501568140575635229180564, −9.684836988140857646197298137617, −8.390725895501417497178439560524, −7.65575386435133920085089846758, −6.75806459593535194272185966722, −5.45631945190590426052679065797, −4.31380434929123365456919576941, −3.19098439098311848143941742422, −1.92527281493145059764635409890, −0.61368356197877171781428678960,
2.66977993955393286922531904956, 3.93206105992000431875377956023, 4.86676653267821044517656487188, 5.92735905989730059813622929160, 6.36425916495185051348726173126, 8.084848028178809405832420785207, 8.662359706554168618312446931876, 9.268106718661234519166811950637, 10.10296072859160192639974015154, 11.51971835604390803014323635727