| L(s) = 1 | + (0.135 + 0.0724i)3-s + (2.12 − 2.59i)5-s + (1.11 + 0.742i)7-s + (−1.65 − 2.47i)9-s + (0.181 + 0.0550i)11-s + (0.374 − 0.307i)13-s + (0.476 − 0.197i)15-s + (1.79 + 0.742i)17-s + (0.487 − 4.94i)19-s + (0.0968 + 0.181i)21-s + (−5.40 − 1.07i)23-s + (−1.22 − 6.14i)25-s + (−0.0900 − 0.914i)27-s + (2.45 + 8.10i)29-s + (−3.30 + 3.30i)31-s + ⋯ |
| L(s) = 1 | + (0.0783 + 0.0418i)3-s + (0.952 − 1.16i)5-s + (0.419 + 0.280i)7-s + (−0.551 − 0.824i)9-s + (0.0547 + 0.0166i)11-s + (0.103 − 0.0851i)13-s + (0.123 − 0.0510i)15-s + (0.434 + 0.180i)17-s + (0.111 − 1.13i)19-s + (0.0211 + 0.0395i)21-s + (−1.12 − 0.224i)23-s + (−0.244 − 1.22i)25-s + (−0.0173 − 0.176i)27-s + (0.456 + 1.50i)29-s + (−0.593 + 0.593i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50171 - 0.795702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50171 - 0.795702i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (-0.135 - 0.0724i)T + (1.66 + 2.49i)T^{2} \) |
| 5 | \( 1 + (-2.12 + 2.59i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.11 - 0.742i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.181 - 0.0550i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-0.374 + 0.307i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-1.79 - 0.742i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.487 + 4.94i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (5.40 + 1.07i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-2.45 - 8.10i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (3.30 - 3.30i)T - 31iT^{2} \) |
| 37 | \( 1 + (-8.32 + 0.820i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.580 + 2.91i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-8.36 + 4.47i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-3.57 + 8.62i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (3.12 - 10.2i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-10.8 - 8.90i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-0.339 + 0.635i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (-0.0570 + 0.106i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (8.81 - 13.1i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (1.93 - 1.29i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-3.11 - 7.51i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (14.1 + 1.39i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (1.57 - 0.313i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (11.0 - 11.0i)T - 97iT^{2} \) |
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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
−10.69018163832551124440752986533, −9.671849003064981139996048971099, −8.904399477292179249738673040309, −8.478839055005259434545500772497, −7.06095199624716430300530185558, −5.83744958653188020272204473317, −5.30171303313423008125713937103, −4.09026153001448460981509610658, −2.54026095931325341443869506667, −1.10092329653888826454355873237,
1.90107641187155588071170328224, 2.88538607045695326639084419311, 4.28755067737302941590630445026, 5.75660314387048805803261375377, 6.19802047269921728315841087055, 7.60461570613809042790232273962, 8.064113193432813495652771754279, 9.586750438461754677681901365019, 10.10039063375868802840903437425, 11.02259421193736523582142962784
