| L(s) = 1 | + (−0.415 − 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (0.654 + 0.755i)6-s + (−0.0949 − 0.660i)7-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.142 + 0.989i)10-s + (−1.84 + 4.02i)11-s + (0.415 − 0.909i)12-s + (0.175 − 1.22i)13-s + (−0.561 + 0.360i)14-s + (0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (2.59 + 3.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (−0.376 − 0.241i)5-s + (0.267 + 0.308i)6-s + (−0.0359 − 0.249i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0450 + 0.313i)10-s + (−0.554 + 1.21i)11-s + (0.119 − 0.262i)12-s + (0.0486 − 0.338i)13-s + (−0.150 + 0.0964i)14-s + (0.247 + 0.0727i)15-s + (−0.0355 − 0.247i)16-s + (0.630 + 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.894583 - 0.217498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.894583 - 0.217498i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (-3.69 + 3.05i)T \) |
| good | 7 | \( 1 + (0.0949 + 0.660i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (1.84 - 4.02i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.175 + 1.22i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.59 - 3.00i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.30 + 2.66i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-4.42 - 5.10i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (8.87 + 2.60i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-4.36 + 2.80i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-8.69 - 5.59i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-11.0 + 3.24i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + (0.349 + 2.42i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.637 - 4.43i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (1.47 + 0.432i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (0.471 + 1.03i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (5.53 + 12.1i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-5.26 + 6.07i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.20 - 8.38i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (13.6 - 8.77i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-3.47 + 1.02i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-9.72 - 6.24i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.72308815290609114273887597704, −9.637070982195063932980591689526, −8.930957704565171374984985944087, −7.68993681908837570383260267046, −7.18113477806202834451543794035, −5.72235009832238238284250033197, −4.76295086337809393597240372603, −3.90738974534299641378338472679, −2.55053446769539464871744003955, −0.934082907829433361617682499751,
0.851368959179158335333875221581, 2.86321336197547702351387013482, 4.18820725922897491750205570480, 5.55949204184549426796485180536, 5.87194699451011659705534452656, 7.21535090274861960685552691280, 7.68138289019464736301179336245, 8.753092644070712905802830277708, 9.552548568325045563434460280224, 10.63370718021332031107781777399
