| L(s) = 1 | + (6.12 + 5.14i)2-s + (10.2 − 3.72i)3-s + (11.1 + 63.0i)4-s + (56.8 − 322. i)5-s + (81.8 + 29.7i)6-s + (−746. − 1.29e3i)7-s + (−256. + 443. i)8-s + (−1.58e3 + 1.32e3i)9-s + (2.00e3 − 1.68e3i)10-s + (2.94e3 − 5.09e3i)11-s + (348. + 603. i)12-s + (−534. − 194. i)13-s + (2.07e3 − 1.17e4i)14-s + (−618. − 3.50e3i)15-s + (−3.84e3 + 1.40e3i)16-s + (−2.09e3 − 1.75e3i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.218 − 0.0796i)3-s + (0.0868 + 0.492i)4-s + (0.203 − 1.15i)5-s + (0.154 + 0.0563i)6-s + (−0.822 − 1.42i)7-s + (−0.176 + 0.306i)8-s + (−0.724 + 0.607i)9-s + (0.634 − 0.532i)10-s + (0.666 − 1.15i)11-s + (0.0582 + 0.100i)12-s + (−0.0674 − 0.0245i)13-s + (0.201 − 1.14i)14-s + (−0.0473 − 0.268i)15-s + (−0.234 + 0.0855i)16-s + (−0.103 − 0.0868i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.60880 - 1.22074i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60880 - 1.22074i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-6.12 - 5.14i)T \) |
| 19 | \( 1 + (-2.71e4 + 1.25e4i)T \) |
| good | 3 | \( 1 + (-10.2 + 3.72i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-56.8 + 322. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (746. + 1.29e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-2.94e3 + 5.09e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (534. + 194. i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (2.09e3 + 1.75e3i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (5.86e3 + 3.32e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (-5.47e4 + 4.59e4i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-7.00e4 - 1.21e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 5.40e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-5.94e5 + 2.16e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (1.71e5 - 9.70e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-1.65e4 + 1.38e4i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (-1.84e4 - 1.04e5i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (1.07e6 + 9.01e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (2.32e5 + 1.32e6i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (-3.13e6 + 2.62e6i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-4.72e5 + 2.68e6i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (-3.50e6 + 1.27e6i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (-2.30e6 + 8.39e5i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-4.95e6 - 8.58e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-3.03e6 - 1.10e6i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (-2.92e6 - 2.45e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−13.97119766003629058091032457565, −13.72063578108408590102214745109, −12.49313323190424097894425448799, −10.96204349676128363978746999643, −9.246865757433637713684079554946, −8.032033869516253421355210900983, −6.48137158689995865969923588540, −4.94257016433580044915607518348, −3.38629636035026718985051004580, −0.72840022236578714977467302241,
2.34157492922988774219741455442, 3.43779415268197315212081486784, 5.70604869204511432838606541276, 6.81972633266268124426771062378, 9.100730604880331320312492793218, 10.04403739913235461618074704247, 11.69156133711386637084818782124, 12.38599631246718590380982878413, 14.02331294979939383959762725990, 14.90563827333396269474659660759
