| L(s) = 1 | − 32i·2-s + 475. i·3-s − 1.02e3·4-s + 1.52e4·6-s − 6.22e3i·7-s + 3.27e4i·8-s − 4.86e4·9-s + 3.83e5·11-s − 4.86e5i·12-s + 3.27e5i·13-s − 1.99e5·14-s + 1.04e6·16-s − 1.06e7i·17-s + 1.55e6i·18-s + 6.77e6·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.12i·3-s − 0.5·4-s + 0.798·6-s − 0.139i·7-s + 0.353i·8-s − 0.274·9-s + 0.717·11-s − 0.564i·12-s + 0.244i·13-s − 0.0989·14-s + 0.250·16-s − 1.81i·17-s + 0.194i·18-s + 0.627·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.51613 + 0.937023i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51613 + 0.937023i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 475. iT - 1.77e5T^{2} \) |
| 7 | \( 1 + 6.22e3iT - 1.97e9T^{2} \) |
| 11 | \( 1 - 3.83e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 3.27e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 1.06e7iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 6.77e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.84e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + 4.02e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.42e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.87e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 8.88e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 8.40e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 1.17e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 - 1.23e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 6.61e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 6.94e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.42e10iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 1.71e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 4.72e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 1.99e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 7.75e9iT - 1.28e21T^{2} \) |
| 89 | \( 1 - 6.32e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.67e10iT - 7.15e21T^{2} \) |
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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
−13.41670938545896538146071295025, −11.83435485937614730491623186957, −11.07679809592179500182594000675, −9.640352747119976477222758179893, −9.299073647941191324940421736605, −7.35619570820090589964003124215, −5.34516658127219180723278494097, −4.20180487566652658495495684370, −3.08742128549138986341501616189, −1.21574795263540872019598168314,
0.59191842832132747201657637092, 1.95252657043239782282373116662, 3.99373798759946033534367622054, 5.82387392044964153429234378725, 6.78033103545357336965464666413, 7.87395332527299899410599189105, 8.964939668753329553799891836080, 10.59810548145624368604212206033, 12.28997745828991302514429174000, 12.90014848019266256630410442546
