| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (0.977 − 0.212i)3-s + (−0.989 − 0.142i)4-s + (2.23 − 0.0804i)5-s + (0.142 + 0.989i)6-s + (−3.44 + 1.87i)7-s + (0.212 − 0.977i)8-s + (0.909 − 0.415i)9-s + (−0.0791 + 2.23i)10-s + (0.512 + 0.444i)11-s + (−0.997 + 0.0713i)12-s + (−3.12 + 5.72i)13-s + (−1.62 − 3.56i)14-s + (2.16 − 0.553i)15-s + (0.959 + 0.281i)16-s + (3.18 + 2.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (0.564 − 0.122i)3-s + (−0.494 − 0.0711i)4-s + (0.999 − 0.0359i)5-s + (0.0580 + 0.404i)6-s + (−1.30 + 0.710i)7-s + (0.0751 − 0.345i)8-s + (0.303 − 0.138i)9-s + (−0.0250 + 0.706i)10-s + (0.154 + 0.133i)11-s + (−0.287 + 0.0205i)12-s + (−0.867 + 1.58i)13-s + (−0.435 − 0.953i)14-s + (0.559 − 0.142i)15-s + (0.239 + 0.0704i)16-s + (0.771 + 0.577i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.938529 + 1.31382i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.938529 + 1.31382i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 3 | \( 1 + (-0.977 + 0.212i)T \) |
| 5 | \( 1 + (-2.23 + 0.0804i)T \) |
| 23 | \( 1 + (1.34 - 4.60i)T \) |
| good | 7 | \( 1 + (3.44 - 1.87i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-0.512 - 0.444i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.12 - 5.72i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-3.18 - 2.38i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.239 - 1.66i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-2.45 + 0.353i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.47 + 3.51i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-4.77 - 1.77i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (2.24 - 4.92i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (2.33 + 10.7i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-4.55 - 4.55i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.23 + 4.09i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (3.74 + 12.7i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-6.18 - 9.62i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (4.78 + 0.342i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (7.81 + 9.01i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (3.48 + 4.64i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (1.48 - 0.435i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (4.16 - 11.1i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-1.40 - 0.900i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (3.39 + 9.09i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.14522212958762986312590790178, −9.616047128193455144113331727767, −9.195783841761094222630114694858, −8.188651289215269972683488022051, −7.02563174742744888359777463192, −6.36036067171364131145471321978, −5.63322673467388962771042222044, −4.34601890122274852315930088844, −3.04700865521298201868914983742, −1.85509654876667139904518334918,
0.827217825511742706887507468909, 2.71428617512312219419924511818, 3.09424606094774700409726954232, 4.50888871796569864063007762017, 5.62586679865339180535696898206, 6.68440035454721354415023373152, 7.67738961088627861540413860193, 8.763451536240792937375414848056, 9.678495113350819471266552733182, 10.15775684489131303325977510100
