| L(s) = 1 | + (−11.0 − 11.0i)3-s + (59.6 + 59.6i)5-s + 660.·7-s + 242. i·9-s + (−767. + 767. i)11-s + (−1.06e3 + 1.06e3i)13-s − 1.31e3i·15-s − 6.83e3·17-s + (−3.47e3 − 3.47e3i)19-s + (−7.28e3 − 7.28e3i)21-s − 8.35e3·23-s − 8.51e3i·25-s + (2.67e3 − 2.67e3i)27-s + (−8.30e3 + 8.30e3i)29-s − 3.56e4i·31-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.476 + 0.476i)5-s + 1.92·7-s + 0.333i·9-s + (−0.576 + 0.576i)11-s + (−0.486 + 0.486i)13-s − 0.389i·15-s − 1.39·17-s + (−0.506 − 0.506i)19-s + (−0.786 − 0.786i)21-s − 0.686·23-s − 0.545i·25-s + (0.136 − 0.136i)27-s + (−0.340 + 0.340i)29-s − 1.19i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.7612753089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7612753089\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (11.0 + 11.0i)T \) |
| good | 5 | \( 1 + (-59.6 - 59.6i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 - 660.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (767. - 767. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (1.06e3 - 1.06e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 6.83e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (3.47e3 + 3.47e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 + 8.35e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (8.30e3 - 8.30e3i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 3.56e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (4.08e4 + 4.08e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 4.89e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (7.79e4 - 7.79e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 8.79e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (1.61e4 + 1.61e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (1.84e4 - 1.84e4i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-2.32e5 + 2.32e5i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (1.09e5 + 1.09e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 6.11e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 4.65e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 8.09e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (7.89e5 + 7.89e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 + 8.37e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.21e5T + 8.32e11T^{2} \) |
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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
−10.22554190678629688052784264308, −8.963248099154310458611516642592, −7.975802877677674548633614561821, −7.20540837832544283929701928065, −6.19761578731852071704100847876, −5.00229161096993337941944193726, −4.39495912473577875545437589486, −2.19534797159929418323254308768, −1.91773243616170935496396574763, −0.16580279140331685576774565771,
1.29005528738701915543596098082, 2.28828669239413866399422314436, 4.03913448385861115609722707480, 5.09973559781700034331618756208, 5.45354580804670990356322920285, 6.90578802823797727744003501443, 8.248914555217395833960112710648, 8.583791163169266119694668887884, 9.936503244000542459201669493060, 10.78105115113489395027318009568
