L(s) = 1 | + (−0.588 − 1.28i)2-s − i·3-s + (−1.30 + 1.51i)4-s + 2.28i·5-s + (−1.28 + 0.588i)6-s + 3.41·7-s + (2.71 + 0.792i)8-s − 9-s + (2.94 − 1.34i)10-s + 1.67·11-s + (1.51 + 1.30i)12-s + 0.215·13-s + (−2.00 − 4.39i)14-s + 2.28·15-s + (−0.577 − 3.95i)16-s + 0.222i·17-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.909i)2-s − 0.577i·3-s + (−0.654 + 0.756i)4-s + 1.02i·5-s + (−0.525 + 0.240i)6-s + 1.29·7-s + (0.959 + 0.280i)8-s − 0.333·9-s + (0.930 − 0.425i)10-s + 0.503·11-s + (0.436 + 0.377i)12-s + 0.0598·13-s + (−0.536 − 1.17i)14-s + 0.590·15-s + (−0.144 − 0.989i)16-s + 0.0540i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03334 - 0.464867i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03334 - 0.464867i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.588 + 1.28i)T \) |
| 3 | \( 1 + iT \) |
| 23 | \( 1 + (4.75 + 0.633i)T \) |
good | 5 | \( 1 - 2.28iT - 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 0.215T + 13T^{2} \) |
| 17 | \( 1 - 0.222iT - 17T^{2} \) |
| 19 | \( 1 - 7.32T + 19T^{2} \) |
| 29 | \( 1 + 6.15T + 29T^{2} \) |
| 31 | \( 1 + 3.33iT - 31T^{2} \) |
| 37 | \( 1 - 1.86iT - 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 - 2.67T + 43T^{2} \) |
| 47 | \( 1 + 6.03iT - 47T^{2} \) |
| 53 | \( 1 - 11.0iT - 53T^{2} \) |
| 59 | \( 1 + 3.93iT - 59T^{2} \) |
| 61 | \( 1 - 9.89iT - 61T^{2} \) |
| 67 | \( 1 + 1.94T + 67T^{2} \) |
| 71 | \( 1 + 8.94iT - 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 4.15T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 11.2iT - 89T^{2} \) |
| 97 | \( 1 + 16.3iT - 97T^{2} \) |
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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
−11.51713858478543105130275532223, −11.10233116319567661173562962962, −10.03099853320347685736258135029, −8.958344478303990830673177627985, −7.80629885033664032381687331769, −7.27646310352248997606664846588, −5.67570756989824282572634038148, −4.16268447404553072570864781722, −2.80118166761162760382864464742, −1.50303478955859304120534783575,
1.34232750092510277097517508901, 4.10739789612718132919226511395, 5.03499250547633439722607078155, 5.74984100537116586064605730576, 7.39433187943319652722742263010, 8.202624497459455906537386601345, 9.063106693977373632940614495417, 9.744310421048105606709748275336, 10.99470399555304080336031053824, 11.84727395596920189767475201016