L(s) = 1 | + (0.540 + 0.540i)2-s + (−0.707 − 0.707i)3-s − 1.41i·4-s + (1.03 − 1.98i)5-s − 0.763i·6-s + (0.614 + 2.57i)7-s + (1.84 − 1.84i)8-s + 1.00i·9-s + (1.63 − 0.510i)10-s − 3.85·11-s + (−1.00 + 1.00i)12-s + (3.66 + 3.66i)13-s + (−1.05 + 1.72i)14-s + (−2.13 + 0.668i)15-s − 0.839·16-s + (−1.49 + 1.49i)17-s + ⋯ |
L(s) = 1 | + (0.381 + 0.381i)2-s + (−0.408 − 0.408i)3-s − 0.708i·4-s + (0.463 − 0.886i)5-s − 0.311i·6-s + (0.232 + 0.972i)7-s + (0.652 − 0.652i)8-s + 0.333i·9-s + (0.515 − 0.161i)10-s − 1.16·11-s + (−0.289 + 0.289i)12-s + (1.01 + 1.01i)13-s + (−0.282 + 0.460i)14-s + (−0.550 + 0.172i)15-s − 0.209·16-s + (−0.361 + 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12765 - 0.275700i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12765 - 0.275700i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.03 + 1.98i)T \) |
| 7 | \( 1 + (-0.614 - 2.57i)T \) |
good | 2 | \( 1 + (-0.540 - 0.540i)T + 2iT^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 + (-3.66 - 3.66i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.49 - 1.49i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.0697T + 19T^{2} \) |
| 23 | \( 1 + (0.534 - 0.534i)T - 23iT^{2} \) |
| 29 | \( 1 - 2.77iT - 29T^{2} \) |
| 31 | \( 1 - 2.39iT - 31T^{2} \) |
| 37 | \( 1 + (-6.18 - 6.18i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.68iT - 41T^{2} \) |
| 43 | \( 1 + (2.77 - 2.77i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.49 - 5.49i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.13 + 6.13i)T - 53iT^{2} \) |
| 59 | \( 1 - 6.97T + 59T^{2} \) |
| 61 | \( 1 + 14.3iT - 61T^{2} \) |
| 67 | \( 1 + (-0.416 - 0.416i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.12T + 71T^{2} \) |
| 73 | \( 1 + (9.55 + 9.55i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.86iT - 79T^{2} \) |
| 83 | \( 1 + (1.63 + 1.63i)T + 83iT^{2} \) |
| 89 | \( 1 - 5.05T + 89T^{2} \) |
| 97 | \( 1 + (6.85 - 6.85i)T - 97iT^{2} \) |
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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
−13.51433653860440575523903572973, −12.95134636219493797674090555660, −11.71745959375170326693700460685, −10.58628121915368912004553743488, −9.299563693317625440963350656639, −8.232001479326197544689929374099, −6.50716401673144048303527600928, −5.62529891424140337140363991638, −4.72765139388627056076311910581, −1.78353771708452073245574297165,
2.84248837881949346888480767133, 4.13993469597125800627486414474, 5.64550737301246195819968457645, 7.18553995974920017023781643621, 8.213993155885694165458740265067, 10.06788912102999941156668609599, 10.80678710274184437418546267218, 11.49068099282408697643337768010, 13.15534128800676918033353602750, 13.43865956838571910387798041195