L(s) = 1 | + (−0.253 − 1.39i)2-s + (−1.87 + 0.704i)4-s + (2.47 + 1.43i)5-s + (2.52 + 0.784i)7-s + (1.45 + 2.42i)8-s + (1.36 − 3.80i)10-s + (0.887 + 1.53i)11-s − 4.70·13-s + (0.451 − 3.71i)14-s + (3.00 − 2.63i)16-s + (−6.08 + 3.51i)17-s + (6.93 + 4.00i)19-s + (−5.64 − 0.930i)20-s + (1.91 − 1.62i)22-s + (−2.46 + 4.26i)23-s + ⋯ |
L(s) = 1 | + (−0.179 − 0.983i)2-s + (−0.935 + 0.352i)4-s + (1.10 + 0.639i)5-s + (0.955 + 0.296i)7-s + (0.514 + 0.857i)8-s + (0.430 − 1.20i)10-s + (0.267 + 0.463i)11-s − 1.30·13-s + (0.120 − 0.992i)14-s + (0.751 − 0.659i)16-s + (−1.47 + 0.852i)17-s + (1.59 + 0.918i)19-s + (−1.26 − 0.208i)20-s + (0.407 − 0.346i)22-s + (−0.513 + 0.889i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.54593 + 0.0948674i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.54593 + 0.0948674i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.253 + 1.39i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.52 - 0.784i)T \) |
good | 5 | \( 1 + (-2.47 - 1.43i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.887 - 1.53i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 + (6.08 - 3.51i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.93 - 4.00i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.46 - 4.26i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.21iT - 29T^{2} \) |
| 31 | \( 1 + (-1.31 + 0.757i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.38 + 2.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.94iT - 41T^{2} \) |
| 43 | \( 1 + 7.39iT - 43T^{2} \) |
| 47 | \( 1 + (-2.78 + 4.82i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.87 - 1.66i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.717 + 1.24i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.30 + 12.6i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.58 - 1.49i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + (-0.0244 - 0.0423i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.15 - 0.668i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.02T + 83T^{2} \) |
| 89 | \( 1 + (5.61 + 3.24i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 1.07T + 97T^{2} \) |
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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
−10.33871654812987893849296964694, −9.677042035270948422309349235279, −8.993019402854208899005713980620, −7.901561954679973980239264671925, −7.00974692390543196371850643625, −5.63101571086482639194246255129, −4.93826007965257743504324150867, −3.69128600216551248538814453867, −2.29467414173042861024423524865, −1.75672630404861499338376858527,
0.888203754673883862491560540248, 2.43657224493445949159833994894, 4.55767549306154347299632310629, 4.89695653953977793694895001954, 5.89310594396397264828903259957, 6.86236883700267143520635373124, 7.71264670688013696766772263522, 8.606114587689764303721866763024, 9.472655773798897624771308099545, 9.827673563605750209548385344630