| L(s) = 1 | − 15.7·3-s + 6.67e3i·5-s − 8.15e4i·7-s − 1.76e5·9-s − 3.04e5i·11-s + (1.14e6 − 6.93e5i)13-s − 1.05e5i·15-s + 6.40e5·17-s − 1.18e7i·19-s + 1.28e6i·21-s + 4.56e7·23-s + 4.24e6·25-s + 5.57e6·27-s + 4.74e6·29-s − 1.52e8i·31-s + ⋯ |
| L(s) = 1 | − 0.0374·3-s + 0.955i·5-s − 1.83i·7-s − 0.998·9-s − 0.569i·11-s + (0.855 − 0.518i)13-s − 0.0357i·15-s + 0.109·17-s − 1.09i·19-s + 0.0686i·21-s + 1.47·23-s + 0.0870·25-s + 0.0747·27-s + 0.0430·29-s − 0.959i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.317724295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.317724295\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (-1.14e6 + 6.93e5i)T \) |
| good | 3 | \( 1 + 15.7T + 1.77e5T^{2} \) |
| 5 | \( 1 - 6.67e3iT - 4.88e7T^{2} \) |
| 7 | \( 1 + 8.15e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 3.04e5iT - 2.85e11T^{2} \) |
| 17 | \( 1 - 6.40e5T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.18e7iT - 1.16e14T^{2} \) |
| 23 | \( 1 - 4.56e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 4.74e6T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.52e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 - 2.70e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 + 9.31e7iT - 5.50e17T^{2} \) |
| 43 | \( 1 + 1.44e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.06e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 - 2.76e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 2.69e9iT - 3.01e19T^{2} \) |
| 61 | \( 1 + 6.51e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 9.67e8iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 1.34e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 8.89e8iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 3.87e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.24e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + 8.88e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 3.90e10iT - 7.15e21T^{2} \) |
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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
−10.34980962961038191845303009814, −8.988355635252405346415098682100, −7.87546855086852633553665081627, −6.96813508607399298594946410424, −6.17019990275398804636364814251, −4.78473689177527059328951715919, −3.46669633803101696747662941723, −2.90936595970338594413645742826, −1.08101181271596242414978509926, −0.28560215659644501384452108422,
1.24778426317118709529413927406, 2.26466950353022771412278108676, 3.42756623157835848356763990287, 5.01352990062198682023156894988, 5.53797236854250548889661035957, 6.60737879618392592862265186116, 8.340257715077951346110485833678, 8.729623760702943614982122807279, 9.491305334081011706179308031340, 10.97574866245149589761215664540
