| L(s) = 1 | + (−2.24 + 8.71i)3-s + (−14.1 − 16.8i)5-s + (35.7 − 13.0i)7-s + (−70.9 − 39.1i)9-s + (−102. + 121. i)11-s + (−49.7 − 282. i)13-s + (178. − 85.6i)15-s + (492. − 284. i)17-s + (135. − 234. i)19-s + (33.2 + 341. i)21-s + (31.0 − 85.3i)23-s + (24.1 − 137. i)25-s + (499. − 530. i)27-s + (−438. − 77.3i)29-s + (1.18e3 + 431. i)31-s + ⋯ |
| L(s) = 1 | + (−0.249 + 0.968i)3-s + (−0.566 − 0.675i)5-s + (0.730 − 0.265i)7-s + (−0.875 − 0.482i)9-s + (−0.846 + 1.00i)11-s + (−0.294 − 1.66i)13-s + (0.795 − 0.380i)15-s + (1.70 − 0.983i)17-s + (0.375 − 0.650i)19-s + (0.0754 + 0.773i)21-s + (0.0587 − 0.161i)23-s + (0.0387 − 0.219i)25-s + (0.685 − 0.727i)27-s + (−0.521 − 0.0920i)29-s + (1.23 + 0.448i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.917i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.396 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.904225 - 0.594280i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.904225 - 0.594280i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.24 - 8.71i)T \) |
| good | 5 | \( 1 + (14.1 + 16.8i)T + (-108. + 615. i)T^{2} \) |
| 7 | \( 1 + (-35.7 + 13.0i)T + (1.83e3 - 1.54e3i)T^{2} \) |
| 11 | \( 1 + (102. - 121. i)T + (-2.54e3 - 1.44e4i)T^{2} \) |
| 13 | \( 1 + (49.7 + 282. i)T + (-2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-492. + 284. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-135. + 234. i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-31.0 + 85.3i)T + (-2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (438. + 77.3i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-1.18e3 - 431. i)T + (7.07e5 + 5.93e5i)T^{2} \) |
| 37 | \( 1 + (522. + 905. i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (2.17e3 - 383. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (2.65e3 + 2.22e3i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (224. + 617. i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 - 4.30e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (1.52e3 + 1.81e3i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-2.63e3 + 958. i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (672. + 3.81e3i)T + (-1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (2.21e3 - 1.28e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-2.21e3 + 3.84e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (985. - 5.58e3i)T + (-3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-3.30e3 - 583. i)T + (4.45e7 + 1.62e7i)T^{2} \) |
| 89 | \( 1 + (2.39e3 + 1.38e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-5.62e3 - 4.72e3i)T + (1.53e7 + 8.71e7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.51811401998487943729901937742, −11.80049436746882238434915777106, −10.49293417545894244093919419171, −9.835157171475715087110932874327, −8.344572857603167231811312572794, −7.52418896987855784305922144854, −5.26414244231463994398226715223, −4.82791287615136255199507098652, −3.14328615372899444346955076993, −0.51876118959095489256256540800,
1.59162279642607664655025816489, 3.29634443705505335838565697254, 5.28582922428532507721624256268, 6.50684940045527536489932544889, 7.73646155259038995297028463853, 8.378280757834818728961182465367, 10.21118630787373580358087739952, 11.56195795642852903541377984557, 11.75287777820980388050096760305, 13.20317586587682479771406882626
