| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−3.09 − 0.270i)3-s + (0.173 + 0.984i)4-s + (1.89 + 1.18i)5-s + (2.19 + 2.19i)6-s + (−1.41 − 3.04i)7-s + (0.500 − 0.866i)8-s + (6.55 + 1.15i)9-s + (−0.685 − 2.12i)10-s + (−3.23 − 1.86i)11-s + (−0.270 − 3.09i)12-s + (0.101 + 0.575i)13-s + (−0.868 + 3.24i)14-s + (−5.54 − 4.19i)15-s + (−0.939 + 0.342i)16-s + (−3.94 − 0.694i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−1.78 − 0.156i)3-s + (0.0868 + 0.492i)4-s + (0.846 + 0.531i)5-s + (0.897 + 0.897i)6-s + (−0.535 − 1.14i)7-s + (0.176 − 0.306i)8-s + (2.18 + 0.385i)9-s + (−0.216 − 0.673i)10-s + (−0.974 − 0.562i)11-s + (−0.0782 − 0.893i)12-s + (0.0281 + 0.159i)13-s + (−0.232 + 0.866i)14-s + (−1.43 − 1.08i)15-s + (−0.234 + 0.0855i)16-s + (−0.955 − 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0847395 + 0.124731i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0847395 + 0.124731i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (-1.89 - 1.18i)T \) |
| 37 | \( 1 + (5.41 + 2.77i)T \) |
| good | 3 | \( 1 + (3.09 + 0.270i)T + (2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (1.41 + 3.04i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (3.23 + 1.86i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.101 - 0.575i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.94 + 0.694i)T + (15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.436 - 4.99i)T + (-18.7 - 3.29i)T^{2} \) |
| 23 | \( 1 + (-3.83 - 6.63i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.76 - 1.27i)T + (25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (0.528 - 0.528i)T - 31iT^{2} \) |
| 41 | \( 1 + (2.10 - 0.371i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + 9.90T + 43T^{2} \) |
| 47 | \( 1 + (3.03 - 11.3i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.25 - 4.84i)T + (-34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (3.84 - 8.24i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (7.23 + 5.06i)T + (20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (-7.11 + 3.31i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-3.02 + 2.54i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.21 - 1.21i)T + 73iT^{2} \) |
| 79 | \( 1 + (-13.9 + 6.51i)T + (50.7 - 60.5i)T^{2} \) |
| 83 | \( 1 + (-6.08 - 8.69i)T + (-28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (9.55 + 4.45i)T + (57.2 + 68.1i)T^{2} \) |
| 97 | \( 1 + (-8.66 + 5.00i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.20139765878985925652128578734, −10.85190691483994120677141965158, −10.20996156049637985895989568284, −9.402408496556928681138719184273, −7.62349938261343753710283601726, −6.85712521780605422874555093892, −6.02112032303839656765439114396, −5.01769925701700740211270414902, −3.48363790103622613272041523782, −1.53233935863331510895234600798,
0.14970095473107033219096190807, 2.17511856197474953904631572875, 4.96863205674932361183894673642, 5.19128685346327348435116364071, 6.35388343210224381988437923703, 6.81697126741137385436612746522, 8.522285648423820597462393559926, 9.403333015566461386901817353750, 10.22991649737437297215922862066, 10.93632405116149095241036006487
