L(s) = 1 | − 2-s + i·3-s + 4-s + (0.5 − 0.866i)5-s − i·6-s + (0.866 + 0.5i)7-s − 8-s − 9-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + i·12-s + (−0.866 − 0.5i)14-s + (0.866 + 0.5i)15-s + 16-s + (−0.5 − 0.866i)17-s + 18-s + ⋯ |
L(s) = 1 | − 2-s + i·3-s + 4-s + (0.5 − 0.866i)5-s − i·6-s + (0.866 + 0.5i)7-s − 8-s − 9-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + i·12-s + (−0.866 − 0.5i)14-s + (0.866 + 0.5i)15-s + 16-s + (−0.5 − 0.866i)17-s + 18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7312163327\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7312163327\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - iT \) |
| 19 | \( 1 + iT \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 - 2iT - T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−10.69909476697934811189131650543, −9.603156136562264448806547697477, −9.048904384568941431311836946424, −8.726049371603263330243804017097, −7.51539214230013025407772844699, −6.35963285201946957935212166092, −5.22771214265526362978252421660, −4.58043641407360458561696458686, −2.91774465881416888625582784135, −1.57799535849528460949180348543,
1.42024011297853700460034194487, 2.34614125849428568897800688149, 3.73828981353941056659928932968, 5.74326736516831024958643843314, 6.43713921962415982824093790586, 7.13232680209401465172906558372, 8.075398854477207882057740293967, 8.607958448817383367216749807969, 9.791646326667150200470554394336, 10.66854960092693126294105270563