L(s) = 1 | − 5.65i·2-s + 14.3i·3-s − 32.0·4-s − 88.1·5-s + 81.2·6-s + 443.·7-s + 181. i·8-s + 522.·9-s + 498. i·10-s + 843.·11-s − 459. i·12-s + 2.24e3i·13-s − 2.50e3i·14-s − 1.26e3i·15-s + 1.02e3·16-s + 8.66e3·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.532i·3-s − 0.500·4-s − 0.705·5-s + 0.376·6-s + 1.29·7-s + 0.353i·8-s + 0.716·9-s + 0.498i·10-s + 0.633·11-s − 0.266i·12-s + 1.02i·13-s − 0.913i·14-s − 0.375i·15-s + 0.250·16-s + 1.76·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.74626 - 0.127159i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.74626 - 0.127159i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 5.65iT \) |
| 19 | \( 1 + (-993. + 6.78e3i)T \) |
good | 3 | \( 1 - 14.3iT - 729T^{2} \) |
| 5 | \( 1 + 88.1T + 1.56e4T^{2} \) |
| 7 | \( 1 - 443.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 843.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.24e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 8.66e3T + 2.41e7T^{2} \) |
| 23 | \( 1 + 4.04e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 1.20e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.29e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 7.14e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.34e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 2.84e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 4.08e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.53e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.87e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.62e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 5.10e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 4.85e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 7.05e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 5.79e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.74e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 4.46e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 5.03e5iT - 8.32e11T^{2} \) |
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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
−14.86255612662393206567381448384, −13.94556064812438904526002107626, −12.15448458825282269555987954404, −11.45348297748849687821185944114, −10.19711817670030501964944273947, −8.853484412730312183070889393435, −7.42106034765480108162797407306, −4.91331465708428092928447319842, −3.80495065365644384940558674838, −1.42248295170733489409069884644,
1.16626919971685974766930555358, 4.06428650327858334560016813089, 5.73528580192971317406601437625, 7.64794114181523467393270731369, 7.983311737315568529823260955623, 9.980592874675244892645845007735, 11.65773543159268229180311709822, 12.66033440895313232367002935533, 14.17881274993847167081262883598, 14.96881238104270612268969927479