| L(s) = 1 | + (−4 + 6.92i)2-s + (20.2 − 35.0i)3-s + (−31.9 − 55.4i)4-s + (−75.0 + 129. i)5-s + (162. + 280. i)6-s − 892.·7-s + 511.·8-s + (272. + 471. i)9-s + (−600. − 1.03e3i)10-s + 7.58e3·11-s − 2.59e3·12-s + (3.99e3 + 6.92e3i)13-s + (3.56e3 − 6.18e3i)14-s + (3.04e3 + 5.26e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (−5.16e3 + 8.95e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.433 − 0.750i)3-s + (−0.249 − 0.433i)4-s + (−0.268 + 0.465i)5-s + (0.306 + 0.530i)6-s − 0.983·7-s + 0.353·8-s + (0.124 + 0.215i)9-s + (−0.189 − 0.328i)10-s + 1.71·11-s − 0.433·12-s + (0.504 + 0.873i)13-s + (0.347 − 0.602i)14-s + (0.232 + 0.403i)15-s + (−0.125 + 0.216i)16-s + (−0.255 + 0.441i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.09842 + 0.890488i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09842 + 0.890488i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (1.22e4 - 2.72e4i)T \) |
| good | 3 | \( 1 + (-20.2 + 35.0i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (75.0 - 129. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + 892.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.58e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-3.99e3 - 6.92e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (5.16e3 - 8.95e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-2.27e4 - 3.94e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (4.59e4 + 7.96e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 2.21e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 9.87e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (4.21e5 - 7.30e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-3.53e5 + 6.12e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (2.75e5 + 4.77e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-2.00e5 - 3.46e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.31e6 + 2.27e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (5.47e5 + 9.48e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.29e6 - 2.24e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.27e6 - 3.94e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-8.10e5 + 1.40e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (2.98e6 - 5.17e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 3.00e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (2.39e6 + 4.14e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.11e6 + 1.93e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−14.95166466382597672041505739116, −13.97930687624686191553778419746, −12.90236556133744143114465171567, −11.41529331258050394712104735363, −9.736458504158734142378893555001, −8.562383360223764400508324874738, −7.07225139856225916462557331471, −6.35455382506626588469076123361, −3.78395349188598448097858573187, −1.54270326184502811225380986547,
0.73998732749255849659750472603, 3.18275310877216646872138879639, 4.32035879997399079194903721594, 6.66892909779603049159532289798, 8.779003306081110535753379888185, 9.348459857180513068285255932748, 10.63453928587539943880991811940, 12.08755719303070786653213595558, 13.09379014961348653876550176533, 14.61808284226574091375283194200
