L(s) = 1 | + (−1.76 + 1.76i)2-s + (−0.743 + 1.56i)3-s − 4.23i·4-s + (−0.269 + 0.269i)5-s + (−1.44 − 4.07i)6-s + (−0.707 + 0.707i)7-s + (3.93 + 3.93i)8-s + (−1.89 − 2.32i)9-s − 0.951i·10-s + (−4.05 − 4.05i)11-s + (6.61 + 3.14i)12-s + (1.60 − 3.22i)13-s − 2.49i·14-s + (−0.221 − 0.622i)15-s − 5.43·16-s + 1.75·17-s + ⋯ |
L(s) = 1 | + (−1.24 + 1.24i)2-s + (−0.429 + 0.903i)3-s − 2.11i·4-s + (−0.120 + 0.120i)5-s + (−0.591 − 1.66i)6-s + (−0.267 + 0.267i)7-s + (1.39 + 1.39i)8-s + (−0.631 − 0.775i)9-s − 0.300i·10-s + (−1.22 − 1.22i)11-s + (1.91 + 0.907i)12-s + (0.445 − 0.895i)13-s − 0.667i·14-s + (−0.0571 − 0.160i)15-s − 1.35·16-s + 0.425·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.203886 - 0.0428215i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.203886 - 0.0428215i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.743 - 1.56i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (-1.60 + 3.22i)T \) |
good | 2 | \( 1 + (1.76 - 1.76i)T - 2iT^{2} \) |
| 5 | \( 1 + (0.269 - 0.269i)T - 5iT^{2} \) |
| 11 | \( 1 + (4.05 + 4.05i)T + 11iT^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 + (2.02 + 2.02i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 29 | \( 1 - 3.90iT - 29T^{2} \) |
| 31 | \( 1 + (2.13 + 2.13i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.60 - 2.60i)T - 37iT^{2} \) |
| 41 | \( 1 + (-5.68 + 5.68i)T - 41iT^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + (7.30 + 7.30i)T + 47iT^{2} \) |
| 53 | \( 1 + 6.27iT - 53T^{2} \) |
| 59 | \( 1 + (7.32 + 7.32i)T + 59iT^{2} \) |
| 61 | \( 1 + 4.57T + 61T^{2} \) |
| 67 | \( 1 + (-9.02 - 9.02i)T + 67iT^{2} \) |
| 71 | \( 1 + (-0.688 + 0.688i)T - 71iT^{2} \) |
| 73 | \( 1 + (7.93 - 7.93i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.01T + 79T^{2} \) |
| 83 | \( 1 + (8.09 - 8.09i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.81 + 3.81i)T + 89iT^{2} \) |
| 97 | \( 1 + (7.04 + 7.04i)T + 97iT^{2} \) |
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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
−11.22250542084184502283149856687, −10.59566623896126209454594582219, −9.853792970806347908791581632101, −8.758236451265926157722108011641, −8.225676485270996847553966012909, −6.98796559957116983659034920137, −5.72065316050783805787667333211, −5.39341618363209756111118073714, −3.32555759365368357671959938550, −0.25104200175606929765447012594,
1.57695398108746821321792168215, 2.69815704764163005436377764673, 4.46009680729594113397764279600, 6.27198356347627719300293588601, 7.59074696941278477676182389520, 8.043870832101199449624619731336, 9.305528174162385822317958589313, 10.23534640860446496913254266903, 10.93167032299020172613249571851, 11.90023045533582719944079716777