| L(s) = 1 | + (−8 + 13.8i)2-s + (29.1 − 50.4i)3-s + (−127. − 221. i)4-s + (−1.35e3 + 2.35e3i)5-s + (465. + 806. i)6-s + 3.58e3·7-s + 4.09e3·8-s + (8.14e3 + 1.41e4i)9-s + (−2.17e4 − 3.76e4i)10-s + 2.03e4·11-s − 1.49e4·12-s + (−6.14e4 − 1.06e5i)13-s + (−2.86e4 + 4.97e4i)14-s + (7.90e4 + 1.36e5i)15-s + (−3.27e4 + 5.67e4i)16-s + (−2.02e5 + 3.50e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.207 − 0.359i)3-s + (−0.249 − 0.433i)4-s + (−0.972 + 1.68i)5-s + (0.146 + 0.254i)6-s + 0.564·7-s + 0.353·8-s + (0.413 + 0.716i)9-s + (−0.687 − 1.19i)10-s + 0.419·11-s − 0.207·12-s + (−0.596 − 1.03i)13-s + (−0.199 + 0.345i)14-s + (0.403 + 0.698i)15-s + (−0.125 + 0.216i)16-s + (−0.587 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.135253 - 0.345515i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.135253 - 0.345515i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8 - 13.8i)T \) |
| 19 | \( 1 + (4.89e5 - 2.88e5i)T \) |
| good | 3 | \( 1 + (-29.1 + 50.4i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.35e3 - 2.35e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 - 3.58e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.03e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (6.14e4 + 1.06e5i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (2.02e5 - 3.50e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 23 | \( 1 + (8.47e5 + 1.46e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (1.13e6 + 1.96e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + 7.59e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.05e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.35e7 + 2.35e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.11e7 - 1.93e7i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-2.05e7 - 3.55e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (1.12e7 + 1.94e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-2.23e7 + 3.87e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (3.80e7 + 6.59e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (3.18e7 + 5.51e7i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (1.76e8 - 3.06e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (1.16e8 - 2.01e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-1.24e8 + 2.15e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 1.03e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-2.35e8 - 4.07e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-7.33e7 + 1.27e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−14.80422115850744078640795220123, −14.46569729239289475650207427883, −12.71702579597082954739726436498, −11.04233742682662276912482256837, −10.35176060052491219172836149525, −8.155411781069197997250157695159, −7.51907138041667265870540593047, −6.30288628175155422284880073760, −4.13498690663358110178619002970, −2.25064012078245250546350156896,
0.14979208083343367507093984319, 1.55022673767558508491343571822, 3.96274184858309079604202690407, 4.75999110977493802085724709485, 7.43604395411551348281717692927, 8.899698397056288386405037347873, 9.363985579948985818587246771713, 11.42125915498825542739393599961, 12.08519915059811986372219649775, 13.22177428386440335137662369783
