| L(s) = 1 | + (9.09 − 6.73i)2-s + (−6.33 + 15.3i)3-s + (37.3 − 122. i)4-s + (−363. + 150. i)5-s + (45.3 + 181. i)6-s + (−912. + 912. i)7-s + (−484. − 1.36e3i)8-s + (1.35e3 + 1.35e3i)9-s + (−2.29e3 + 3.82e3i)10-s + (−562. − 1.35e3i)11-s + (1.63e3 + 1.34e3i)12-s + (−1.00e4 − 4.18e3i)13-s + (−2.15e3 + 1.44e4i)14-s − 6.52e3i·15-s + (−1.35e4 − 9.14e3i)16-s + 2.38e4i·17-s + ⋯ |
| L(s) = 1 | + (0.803 − 0.595i)2-s + (−0.135 + 0.327i)3-s + (0.291 − 0.956i)4-s + (−1.30 + 0.539i)5-s + (0.0857 + 0.343i)6-s + (−1.00 + 1.00i)7-s + (−0.334 − 0.942i)8-s + (0.618 + 0.618i)9-s + (−0.725 + 1.20i)10-s + (−0.127 − 0.307i)11-s + (0.273 + 0.225i)12-s + (−1.27 − 0.527i)13-s + (−0.209 + 1.40i)14-s − 0.498i·15-s + (−0.829 − 0.558i)16-s + 1.17i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.271499 + 0.536676i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.271499 + 0.536676i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-9.09 + 6.73i)T \) |
| good | 3 | \( 1 + (6.33 - 15.3i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (363. - 150. i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (912. - 912. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (562. + 1.35e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (1.00e4 + 4.18e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 2.38e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-2.42e4 - 1.00e4i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (2.67e4 + 2.67e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (-5.26e4 + 1.27e5i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 2.65e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-5.04e5 + 2.08e5i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (1.98e5 + 1.98e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (-1.16e5 - 2.80e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 8.25e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-5.73e5 - 1.38e6i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (1.52e6 - 6.31e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-4.49e5 + 1.08e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (1.17e6 - 2.82e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (2.09e6 - 2.09e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (-1.13e6 - 1.13e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 - 9.34e4iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (7.26e5 + 3.00e5i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (1.43e6 - 1.43e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 - 1.07e6T + 8.07e13T^{2} \) |
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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
−15.49941679289547658716574601175, −14.68921948980189206199511949547, −12.87595981801980641289459409939, −12.11670042462140097044234601121, −10.85559559794724987479876773414, −9.723030792837263152955290211160, −7.56258554152607806154988347550, −5.80453928607843686525109939088, −4.10333233235591084864742439847, −2.72673075625870890549470811337,
0.21195231624101734728539091625, 3.51993960436311815241736395733, 4.73181283973750569115879247020, 6.99405930565713244735637241323, 7.47074392579231082861639227714, 9.520699955110706424656804790281, 11.69438098840680152172143444005, 12.45705873829991697193890803514, 13.49415500731690609038441069799, 14.95582298364713105037945275750
