| L(s) = 1 | + (4 − 6.92i)2-s + (9.55 − 16.5i)3-s + (−31.9 − 55.4i)4-s + (226. − 391. i)5-s + (−76.4 − 132. i)6-s − 488.·7-s − 511.·8-s + (910. + 1.57e3i)9-s + (−1.80e3 − 3.13e3i)10-s + 1.81e3·11-s − 1.22e3·12-s + (−7.65e3 − 1.32e4i)13-s + (−1.95e3 + 3.38e3i)14-s + (−4.31e3 − 7.48e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (−2.17e3 + 3.76e3i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.204 − 0.353i)3-s + (−0.249 − 0.433i)4-s + (0.808 − 1.40i)5-s + (−0.144 − 0.250i)6-s − 0.538·7-s − 0.353·8-s + (0.416 + 0.721i)9-s + (−0.571 − 0.990i)10-s + 0.410·11-s − 0.204·12-s + (−0.966 − 1.67i)13-s + (−0.190 + 0.329i)14-s + (−0.330 − 0.572i)15-s + (−0.125 + 0.216i)16-s + (−0.107 + 0.185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.587446 - 2.07576i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.587446 - 2.07576i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 + 6.92i)T \) |
| 19 | \( 1 + (2.81e4 - 9.93e3i)T \) |
| good | 3 | \( 1 + (-9.55 + 16.5i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-226. + 391. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + 488.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.81e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (7.65e3 + 1.32e4i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (2.17e3 - 3.76e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-7.98e3 - 1.38e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (3.91e4 + 6.78e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.36e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.68e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.71e5 + 2.96e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-2.22e5 + 3.86e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-5.07e5 - 8.79e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-8.71e5 - 1.50e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-8.24e5 + 1.42e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (8.46e4 + 1.46e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.52e6 + 2.63e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-9.53e5 + 1.65e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.00e6 + 1.73e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (2.92e6 - 5.07e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 4.58e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-1.61e6 - 2.79e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (3.74e4 - 6.49e4i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.81091337852774990792477363878, −12.87537375728299835622644849963, −12.45436813867761120880952262594, −10.43195357999195294796152496143, −9.436492244294710284756734425486, −8.014354284892134327109220185363, −5.87290355956344126691205350396, −4.59519242423417075182167514766, −2.37453771521164405897726090966, −0.845731616728194843741934547155,
2.60575885286186021840494555319, 4.23186210577492710035796356683, 6.41548956486174002305484851734, 6.91397226631878448197304978922, 9.196428629469948944069353973132, 10.05773552282161016957921504468, 11.69110480393224695195922422828, 13.22949274350316158166774368379, 14.50974951517469667296456945209, 14.83824156178788286164287447800
