L(s) = 1 | + (−1.73 + i)2-s + (0.999 − 1.73i)4-s + (2.23 − 0.133i)5-s + 4i·7-s + (−1.5 + 2.59i)9-s + (−3.73 + 2.46i)10-s − 11-s + (1.73 + i)13-s + (−4 − 6.92i)14-s + (1.99 + 3.46i)16-s + (1.73 − i)17-s − 6i·18-s + (3.5 − 2.59i)19-s + (1.99 − 3.99i)20-s + (1.73 − i)22-s + (−5.19 − 3i)23-s + ⋯ |
L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.499 − 0.866i)4-s + (0.998 − 0.0599i)5-s + 1.51i·7-s + (−0.5 + 0.866i)9-s + (−1.18 + 0.779i)10-s − 0.301·11-s + (0.480 + 0.277i)13-s + (−1.06 − 1.85i)14-s + (0.499 + 0.866i)16-s + (0.420 − 0.242i)17-s − 1.41i·18-s + (0.802 − 0.596i)19-s + (0.447 − 0.894i)20-s + (0.369 − 0.213i)22-s + (−1.08 − 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0379 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0379 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.445300 + 0.428717i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.445300 + 0.428717i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.23 + 0.133i)T \) |
| 19 | \( 1 + (-3.5 + 2.59i)T \) |
good | 2 | \( 1 + (1.73 - i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.73 + i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.19 - 3i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.46 + 2i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.66 + 5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.66 + 5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (5.5 - 9.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \) |
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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
−14.40947868630918983489018495346, −13.34982845221011465360419029984, −12.04858975361556599237017375013, −10.65305410556620146431438394270, −9.532039439246845229554027544587, −8.806519243418086288951730869509, −7.83893619798098288182257797829, −6.24988958210879402343157399272, −5.40320872327279133319137090296, −2.32251101184469232458412983922,
1.27418545886597354390154600307, 3.37630375747422206201856635784, 5.66588674590193468099192679184, 7.22009183458066532805278103219, 8.479769171669099165672302364040, 9.679195793232533285153980504497, 10.28406807025431810497662859055, 11.16431003051782320385362028852, 12.48437460523394184408603202470, 13.81946241582547395155038036360