L(s) = 1 | + 13.8i·3-s − 18.1·5-s + 85.5·7-s + 536.·9-s − 316.·11-s + 2.45e3i·13-s − 251. i·15-s − 3.22e3·17-s + (2.50e3 + 6.38e3i)19-s + 1.18e3i·21-s − 7.82e3·23-s − 1.52e4·25-s + 1.75e4i·27-s + 1.89e4i·29-s + 3.23e4i·31-s + ⋯ |
L(s) = 1 | + 0.514i·3-s − 0.144·5-s + 0.249·7-s + 0.735·9-s − 0.237·11-s + 1.11i·13-s − 0.0744i·15-s − 0.656·17-s + (0.364 + 0.931i)19-s + 0.128i·21-s − 0.643·23-s − 0.978·25-s + 0.892i·27-s + 0.778i·29-s + 1.08i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.836896 + 1.22692i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.836896 + 1.22692i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.50e3 - 6.38e3i)T \) |
good | 3 | \( 1 - 13.8iT - 729T^{2} \) |
| 5 | \( 1 + 18.1T + 1.56e4T^{2} \) |
| 7 | \( 1 - 85.5T + 1.17e5T^{2} \) |
| 11 | \( 1 + 316.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.45e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 3.22e3T + 2.41e7T^{2} \) |
| 23 | \( 1 + 7.82e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 1.89e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.23e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 1.54e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 9.41e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 3.92e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 9.23e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.30e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.80e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.24e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.54e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 1.77e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.14e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.62e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 6.01e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 2.75e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.52e5iT - 8.32e11T^{2} \) |
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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
−13.73565509210489010790066809131, −12.47162382741404136690591056742, −11.38352828330163779999212674280, −10.23730585214808248687528911700, −9.254478835036504914968474841988, −7.890083786603145152147603804538, −6.56865636850394306210888730745, −4.90187647161800223428812063302, −3.76795837058055505795475155777, −1.71789443610224450913029908938,
0.56813098159546206864079020673, 2.30072435624183687155066593109, 4.18133599241282527426623013226, 5.75241330727500396079620830143, 7.21877481102506702533306406393, 8.099649026107955934554860185292, 9.605661706029356167141268217963, 10.78093167145188827221952823243, 11.96875303884928990701082475318, 13.02987405319089056447255992781