| L(s) = 1 | + (−0.762 + 1.84i)2-s + (−1.47 + 0.908i)3-s + (−1.39 − 1.39i)4-s + (−0.360 − 0.240i)5-s + (−0.548 − 3.40i)6-s + (1.52 + 2.28i)7-s + (−0.0550 + 0.0227i)8-s + (1.34 − 2.68i)9-s + (0.717 − 0.479i)10-s + (5.59 + 1.11i)11-s + (3.32 + 0.787i)12-s + (−1.33 + 1.33i)13-s + (−5.38 + 1.07i)14-s + (0.749 + 0.0274i)15-s − 4.05i·16-s + (−0.113 − 4.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.539 + 1.30i)2-s + (−0.851 + 0.524i)3-s + (−0.696 − 0.696i)4-s + (−0.161 − 0.107i)5-s + (−0.224 − 1.39i)6-s + (0.578 + 0.865i)7-s + (−0.0194 + 0.00805i)8-s + (0.449 − 0.893i)9-s + (0.226 − 0.151i)10-s + (1.68 + 0.335i)11-s + (0.958 + 0.227i)12-s + (−0.370 + 0.370i)13-s + (−1.43 + 0.286i)14-s + (0.193 + 0.00709i)15-s − 1.01i·16-s + (−0.0274 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.206422 + 0.498985i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.206422 + 0.498985i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.47 - 0.908i)T \) |
| 17 | \( 1 + (0.113 + 4.12i)T \) |
| good | 2 | \( 1 + (0.762 - 1.84i)T + (-1.41 - 1.41i)T^{2} \) |
| 5 | \( 1 + (0.360 + 0.240i)T + (1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-1.52 - 2.28i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-5.59 - 1.11i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (1.33 - 1.33i)T - 13iT^{2} \) |
| 19 | \( 1 + (0.697 + 0.288i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.356 + 1.79i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (3.19 - 4.78i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + (0.0104 + 0.0525i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (-8.50 + 1.69i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.34 + 0.901i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (3.22 - 1.33i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (1.44 + 1.44i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.67 + 6.46i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (8.58 - 3.55i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (7.75 - 5.17i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 - 2.50iT - 67T^{2} \) |
| 71 | \( 1 + (-1.26 - 6.36i)T + (-65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.785 + 1.17i)T + (-27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-1.89 + 9.53i)T + (-72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (7.32 + 3.03i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-7.83 + 7.83i)T - 89iT^{2} \) |
| 97 | \( 1 + (9.31 + 6.22i)T + (37.1 + 89.6i)T^{2} \) |
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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
−16.19695981548189391585142273630, −15.02971059142943585847438858402, −14.45659197329088279044889529655, −12.13975723185710780905088578728, −11.51650247514923135079506016852, −9.582657491686468883975807950055, −8.795443790020237025275988355331, −7.11173654432139022051016898951, −6.02522185593360054869163433426, −4.66112471116452129310698598716,
1.39455733149245546017578687276, 4.02416664487946296769324408827, 6.26645958929109366493434958162, 7.83458508998544597944525266983, 9.522747169675909939233375879397, 10.81394216522901071206496849870, 11.41731589832778237389859288701, 12.36850352684301192724639868333, 13.56157756776337786443817336715, 15.01718103886996666729463857848
