L(s) = 1 | + 43.2i·3-s − 100.·5-s − 640.·7-s − 1.14e3·9-s + 937.·11-s − 3.44e3i·13-s − 4.34e3i·15-s − 5.12e3·17-s + (−4.65e3 + 5.03e3i)19-s − 2.77e4i·21-s + 1.58e4·23-s − 5.54e3·25-s − 1.79e4i·27-s + 2.40e3i·29-s − 2.27e4i·31-s + ⋯ |
L(s) = 1 | + 1.60i·3-s − 0.803·5-s − 1.86·7-s − 1.56·9-s + 0.704·11-s − 1.56i·13-s − 1.28i·15-s − 1.04·17-s + (−0.678 + 0.734i)19-s − 2.99i·21-s + 1.30·23-s − 0.354·25-s − 0.912i·27-s + 0.0985i·29-s − 0.762i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.6327797137\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6327797137\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (4.65e3 - 5.03e3i)T \) |
good | 3 | \( 1 - 43.2iT - 729T^{2} \) |
| 5 | \( 1 + 100.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 640.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 937.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 3.44e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 5.12e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 1.58e4T + 1.48e8T^{2} \) |
| 29 | \( 1 - 2.40e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.27e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 8.65e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 9.00e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.22e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 4.32e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 3.73e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.69e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.49e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.11e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.69e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.94e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.22e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 5.37e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 3.78e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.56e6iT - 8.32e11T^{2} \) |
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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
−10.60104350432726814660978873628, −9.886268140343788106747274880721, −9.194747761072663173104312073330, −8.222929533643111811121227026711, −6.76371393222602410796127258170, −5.78971569196682904846983195220, −4.49140841350190065715116317844, −3.58250788943493575657715504943, −3.01865069755085197502915766602, −0.32803704574403670605238572794,
0.47569613745937485868508921972, 1.89148002800404195888031399075, 3.09480106277494280475310345824, 4.28159430504166827681438436666, 6.22050097067745533255600648698, 6.82629933795369243749385856390, 7.20473397771357216687003106320, 8.786414931369977884893273452329, 9.206249951703595155755229801098, 10.81898251990225027300209310016