| L(s) = 1 | + (5.19 − 4.35i)5-s + (−17.8 + 6.47i)7-s + (0.134 + 0.112i)11-s + (−1.24 − 7.04i)13-s + (26.6 + 46.0i)17-s + (−65.4 + 113. i)19-s + (−129. − 46.9i)23-s + (−13.7 + 77.8i)25-s + (9.32 − 52.8i)29-s + (139. + 50.6i)31-s + (−64.2 + 111. i)35-s + (58.6 + 101. i)37-s + (27.9 + 158. i)41-s + (51.8 + 43.5i)43-s + (−597. + 217. i)47-s + ⋯ |
| L(s) = 1 | + (0.464 − 0.389i)5-s + (−0.961 + 0.349i)7-s + (0.00367 + 0.00308i)11-s + (−0.0265 − 0.150i)13-s + (0.379 + 0.657i)17-s + (−0.790 + 1.36i)19-s + (−1.17 − 0.426i)23-s + (−0.109 + 0.622i)25-s + (0.0597 − 0.338i)29-s + (0.806 + 0.293i)31-s + (−0.310 + 0.537i)35-s + (0.260 + 0.451i)37-s + (0.106 + 0.603i)41-s + (0.184 + 0.154i)43-s + (−1.85 + 0.675i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8681760794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8681760794\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-5.19 + 4.35i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (17.8 - 6.47i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-0.134 - 0.112i)T + (231. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (1.24 + 7.04i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-26.6 - 46.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (65.4 - 113. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (129. + 46.9i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-9.32 + 52.8i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-139. - 50.6i)T + (2.28e4 + 1.91e4i)T^{2} \) |
| 37 | \( 1 + (-58.6 - 101. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-27.9 - 158. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-51.8 - 43.5i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (597. - 217. i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + 36.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + (574. - 482. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (64.9 - 23.6i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-109. - 622. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-66.9 - 116. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (435. - 754. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-55.7 + 315. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-197. + 1.12e3i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-259. + 449. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.15e3 + 968. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−11.61848446628594056965512793984, −10.22229869191921732440017661569, −9.833632904734979160071030433117, −8.684425892760946965023596284301, −7.83704976353799798683810762883, −6.30675799398417530365360420052, −5.86372249109228725542741885744, −4.37727050900417699692291805710, −3.10732565678725641435196398848, −1.62325297323384437455900161703,
0.29929417713492779046989521025, 2.27432105401489084685934567928, 3.45968233542158457718191412411, 4.79277162624283957502531985706, 6.19192816743620985553190853735, 6.78972205623952955376189611652, 7.956914549031210074374552505060, 9.238654449977459335210400532396, 9.906385361199367085135746405606, 10.74718695406267888055535325246
