L(s) = 1 | + (0.707 − 0.707i)3-s + (1.76 − 1.76i)5-s + (−0.636 + 2.56i)7-s − 1.00i·9-s + (−0.930 + 0.930i)11-s + (0.170 + 0.170i)13-s − 2.49i·15-s + 4.42i·17-s + (4.04 − 4.04i)19-s + (1.36 + 2.26i)21-s + 9.00·23-s − 1.22i·25-s + (−0.707 − 0.707i)27-s + (7.38 − 7.38i)29-s + 3.11·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.789 − 0.789i)5-s + (−0.240 + 0.970i)7-s − 0.333i·9-s + (−0.280 + 0.280i)11-s + (0.0473 + 0.0473i)13-s − 0.644i·15-s + 1.07i·17-s + (0.928 − 0.928i)19-s + (0.298 + 0.494i)21-s + 1.87·23-s − 0.245i·25-s + (−0.136 − 0.136i)27-s + (1.37 − 1.37i)29-s + 0.559·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.254535102\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.254535102\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 + (0.636 - 2.56i)T \) |
good | 5 | \( 1 + (-1.76 + 1.76i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.930 - 0.930i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.170 - 0.170i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.42iT - 17T^{2} \) |
| 19 | \( 1 + (-4.04 + 4.04i)T - 19iT^{2} \) |
| 23 | \( 1 - 9.00T + 23T^{2} \) |
| 29 | \( 1 + (-7.38 + 7.38i)T - 29iT^{2} \) |
| 31 | \( 1 - 3.11T + 31T^{2} \) |
| 37 | \( 1 + (2.72 + 2.72i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.02T + 41T^{2} \) |
| 43 | \( 1 + (-2.03 + 2.03i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.64T + 47T^{2} \) |
| 53 | \( 1 + (-3.39 - 3.39i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.18 + 2.18i)T + 59iT^{2} \) |
| 61 | \( 1 + (4.58 + 4.58i)T + 61iT^{2} \) |
| 67 | \( 1 + (-6.50 - 6.50i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.26T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 + 6.70iT - 79T^{2} \) |
| 83 | \( 1 + (7.18 - 7.18i)T - 83iT^{2} \) |
| 89 | \( 1 - 5.14T + 89T^{2} \) |
| 97 | \( 1 - 2.08iT - 97T^{2} \) |
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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
−9.370090980651297788650840166416, −8.810504965166312242098915026865, −8.148652420458672732924117319276, −7.05129894311482451021735455870, −6.20144953986994692574044130493, −5.38611847046095548675293936882, −4.63494508325299962778180892391, −3.12187472347103648764569116548, −2.28468263813098557180144240691, −1.12294267162694476018401289546,
1.21528800291741534757688519829, 2.89173577871235915244854268383, 3.22877668785646088434973153969, 4.62128200476154254048666976791, 5.40843341015533148773966648050, 6.60258812369253437432615796913, 7.10551633671502810961090345246, 8.038937436513051396115037565778, 9.043667018489798758520177944924, 9.813655152816319197628318598574