L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.987 + 1.71i)3-s + (−0.499 + 0.866i)4-s + (1.00 − 1.99i)5-s + 1.97·6-s + (1.58 + 2.12i)7-s + 0.999·8-s + (−0.451 − 0.782i)9-s + (−2.23 + 0.127i)10-s + (1.14 − 3.11i)11-s + (−0.987 − 1.71i)12-s + 4.36i·13-s + (1.04 − 2.42i)14-s + (2.42 + 3.69i)15-s + (−0.5 − 0.866i)16-s + (−0.154 − 0.0893i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.570 + 0.987i)3-s + (−0.249 + 0.433i)4-s + (0.449 − 0.893i)5-s + 0.806·6-s + (0.597 + 0.802i)7-s + 0.353·8-s + (−0.150 − 0.260i)9-s + (−0.705 + 0.0404i)10-s + (0.344 − 0.938i)11-s + (−0.285 − 0.493i)12-s + 1.21i·13-s + (0.280 − 0.649i)14-s + (0.625 + 0.953i)15-s + (−0.125 − 0.216i)16-s + (−0.0375 − 0.0216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16686 + 0.280478i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16686 + 0.280478i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-1.00 + 1.99i)T \) |
| 7 | \( 1 + (-1.58 - 2.12i)T \) |
| 11 | \( 1 + (-1.14 + 3.11i)T \) |
good | 3 | \( 1 + (0.987 - 1.71i)T + (-1.5 - 2.59i)T^{2} \) |
| 13 | \( 1 - 4.36iT - 13T^{2} \) |
| 17 | \( 1 + (0.154 + 0.0893i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.57 + 2.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.01 + 3.47i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.86iT - 29T^{2} \) |
| 31 | \( 1 + (-6.35 - 3.67i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.87 - 1.08i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.35T + 41T^{2} \) |
| 43 | \( 1 + 1.81T + 43T^{2} \) |
| 47 | \( 1 + (-1.99 - 3.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.2 - 5.91i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.30 + 3.63i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.80 - 13.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.07 - 2.92i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.02T + 71T^{2} \) |
| 73 | \( 1 + (4.65 + 2.68i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.13 + 2.96i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.33iT - 83T^{2} \) |
| 89 | \( 1 + (5.10 - 2.94i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.52T + 97T^{2} \) |
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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
−10.54032328127146595129303374294, −9.373696592440334845396634145877, −8.955257735179269652885566786066, −8.378900175639631584690665151420, −6.78988634501766646130314668788, −5.61572572575601249279651864504, −4.84540584513382121753698818071, −4.20058731647363194881973899436, −2.64318389229700697882582147667, −1.25039975205816299294048533538,
0.898162557416265785265574205220, 2.16009791808413484164307957248, 3.87786608780479268274838916781, 5.22655959885012323303488937505, 6.13324094238148161377470579299, 6.86474240067768033556122067597, 7.50240825916811640419614050222, 8.092277924120831300964503856447, 9.591063076098597251367125767184, 10.18389404528317147738822574333