| L(s) = 1 | + (−0.173 + 0.984i)2-s + (1.32 + 0.928i)3-s + (−0.939 − 0.342i)4-s + (0.101 − 2.23i)5-s + (−1.14 + 1.14i)6-s + (0.111 − 1.27i)7-s + (0.5 − 0.866i)8-s + (−0.130 − 0.358i)9-s + (2.18 + 0.487i)10-s + (3.41 + 1.97i)11-s + (−0.928 − 1.32i)12-s + (5.67 + 2.06i)13-s + (1.23 + 0.331i)14-s + (2.20 − 2.86i)15-s + (0.766 + 0.642i)16-s + (1.63 + 4.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.765 + 0.535i)3-s + (−0.469 − 0.171i)4-s + (0.0453 − 0.998i)5-s + (−0.467 + 0.467i)6-s + (0.0421 − 0.481i)7-s + (0.176 − 0.306i)8-s + (−0.0435 − 0.119i)9-s + (0.690 + 0.154i)10-s + (1.02 + 0.594i)11-s + (−0.267 − 0.382i)12-s + (1.57 + 0.572i)13-s + (0.330 + 0.0885i)14-s + (0.570 − 0.740i)15-s + (0.191 + 0.160i)16-s + (0.397 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56611 + 0.545227i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56611 + 0.545227i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 5 | \( 1 + (-0.101 + 2.23i)T \) |
| 37 | \( 1 + (0.431 + 6.06i)T \) |
| good | 3 | \( 1 + (-1.32 - 0.928i)T + (1.02 + 2.81i)T^{2} \) |
| 7 | \( 1 + (-0.111 + 1.27i)T + (-6.89 - 1.21i)T^{2} \) |
| 11 | \( 1 + (-3.41 - 1.97i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.67 - 2.06i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.63 - 4.50i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (1.17 - 1.67i)T + (-6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (2.61 + 4.52i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.104 - 0.389i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (3.76 + 3.76i)T + 31iT^{2} \) |
| 41 | \( 1 + (3.22 - 8.85i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 - 2.78T + 43T^{2} \) |
| 47 | \( 1 + (10.8 + 2.89i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.144 + 1.65i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (0.316 + 3.61i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (-3.38 - 7.25i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-2.47 - 0.216i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-2.29 - 13.0i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (8.34 - 8.34i)T - 73iT^{2} \) |
| 79 | \( 1 + (7.48 + 0.654i)T + (77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (1.02 + 0.475i)T + (53.3 + 63.5i)T^{2} \) |
| 89 | \( 1 + (7.22 - 0.631i)T + (87.6 - 15.4i)T^{2} \) |
| 97 | \( 1 + (-9.95 + 5.74i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.50037037777832173152612920045, −10.19253731448949451558297063099, −9.408862590856157358484409961625, −8.618120950247527072780965041959, −8.155397831565729320137316209609, −6.66799360069664487256057794008, −5.81777497672384489930527659224, −4.08548005809323206398056002297, −4.00847639361745007961549927363, −1.46190433008559258021937067370,
1.61919721048082126541366324109, 2.96482880771565370958080795476, 3.62060357641107092006782445767, 5.51893938275832068357918270374, 6.63713099294158067657698313798, 7.74494152519306199337478214386, 8.621669123518357080713598367010, 9.350356093043924426899582694062, 10.57090453388373112647567337783, 11.29142707884230436901025349462
