L(s) = 1 | − 17.1i·3-s + 166.·5-s + 600.·7-s + 434.·9-s − 821.·11-s − 3.12e3i·13-s − 2.86e3i·15-s + 6.60e3·17-s + (−2.66e3 − 6.32e3i)19-s − 1.02e4i·21-s + 3.41e3·23-s + 1.22e4·25-s − 1.99e4i·27-s − 3.63e4i·29-s + 3.83e4i·31-s + ⋯ |
L(s) = 1 | − 0.635i·3-s + 1.33·5-s + 1.75·7-s + 0.596·9-s − 0.616·11-s − 1.42i·13-s − 0.848i·15-s + 1.34·17-s + (−0.388 − 0.921i)19-s − 1.11i·21-s + 0.280·23-s + 0.784·25-s − 1.01i·27-s − 1.49i·29-s + 1.28i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(3.852545308\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.852545308\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.66e3 + 6.32e3i)T \) |
good | 3 | \( 1 + 17.1iT - 729T^{2} \) |
| 5 | \( 1 - 166.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 600.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 821.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 3.12e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.60e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 3.41e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 3.63e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.83e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 7.14e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 7.66e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.28e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.00e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 1.51e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 8.63e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.48e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.36e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 6.80e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.79e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.13e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 3.50e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 4.59e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.48e5iT - 8.32e11T^{2} \) |
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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
−10.37773441285563785432055137144, −9.825145753310576950152787005460, −8.242978576590373318130780831056, −7.85909764975728223355696802398, −6.58659188074981069870619274654, −5.41342944922625121052420744830, −4.83569234499008970373670804319, −2.82109163208456474767689119446, −1.71091968735778893312148426780, −0.982592117501664767587316841423,
1.47958124785640788940539408416, 2.00841267695890335547254485666, 3.88181195930792655158055907706, 4.99439112123073673744342965829, 5.57933895964169255219486159886, 7.01600235819030042609416345940, 8.096475662313571198494937212176, 9.147539580604141383167461369703, 9.999007367650930243704513839023, 10.66492030593117238551864979595