| L(s) = 1 | + (−15.1 + 9.71i)2-s + (3.84 − 26.7i)3-s + (81.0 − 177. i)4-s + (−131. + 38.5i)5-s + (201. + 441. i)6-s + (−595. − 687. i)7-s + (171. + 1.19e3i)8-s + (−699. − 205. i)9-s + (1.61e3 − 1.85e3i)10-s + (−6.51e3 − 4.18e3i)11-s + (−4.43e3 − 2.84e3i)12-s + (4.09e3 − 4.72e3i)13-s + (1.56e4 + 4.60e3i)14-s + (525. + 3.65e3i)15-s + (2.16e3 + 2.49e3i)16-s + (1.13e4 + 2.48e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.33 + 0.858i)2-s + (0.0821 − 0.571i)3-s + (0.633 − 1.38i)4-s + (−0.469 + 0.137i)5-s + (0.381 + 0.834i)6-s + (−0.656 − 0.757i)7-s + (0.118 + 0.824i)8-s + (−0.319 − 0.0939i)9-s + (0.509 − 0.587i)10-s + (−1.47 − 0.948i)11-s + (−0.740 − 0.475i)12-s + (0.516 − 0.596i)13-s + (1.52 + 0.448i)14-s + (0.0402 + 0.279i)15-s + (0.131 + 0.152i)16-s + (0.559 + 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0516498 + 0.107595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0516498 + 0.107595i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.84 + 26.7i)T \) |
| 23 | \( 1 + (2.43e4 + 5.30e4i)T \) |
| good | 2 | \( 1 + (15.1 - 9.71i)T + (53.1 - 116. i)T^{2} \) |
| 5 | \( 1 + (131. - 38.5i)T + (6.57e4 - 4.22e4i)T^{2} \) |
| 7 | \( 1 + (595. + 687. i)T + (-1.17e5 + 8.15e5i)T^{2} \) |
| 11 | \( 1 + (6.51e3 + 4.18e3i)T + (8.09e6 + 1.77e7i)T^{2} \) |
| 13 | \( 1 + (-4.09e3 + 4.72e3i)T + (-8.93e6 - 6.21e7i)T^{2} \) |
| 17 | \( 1 + (-1.13e4 - 2.48e4i)T + (-2.68e8 + 3.10e8i)T^{2} \) |
| 19 | \( 1 + (-2.02e4 + 4.43e4i)T + (-5.85e8 - 6.75e8i)T^{2} \) |
| 29 | \( 1 + (-8.87e4 - 1.94e5i)T + (-1.12e10 + 1.30e10i)T^{2} \) |
| 31 | \( 1 + (1.59e4 + 1.10e5i)T + (-2.63e10 + 7.75e9i)T^{2} \) |
| 37 | \( 1 + (3.63e5 + 1.06e5i)T + (7.98e10 + 5.13e10i)T^{2} \) |
| 41 | \( 1 + (3.61e5 - 1.06e5i)T + (1.63e11 - 1.05e11i)T^{2} \) |
| 43 | \( 1 + (1.11e5 - 7.73e5i)T + (-2.60e11 - 7.65e10i)T^{2} \) |
| 47 | \( 1 - 2.87e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-8.91e5 - 1.02e6i)T + (-1.67e11 + 1.16e12i)T^{2} \) |
| 59 | \( 1 + (9.92e5 - 1.14e6i)T + (-3.54e11 - 2.46e12i)T^{2} \) |
| 61 | \( 1 + (-1.68e5 - 1.16e6i)T + (-3.01e12 + 8.85e11i)T^{2} \) |
| 67 | \( 1 + (-1.38e6 + 8.91e5i)T + (2.51e12 - 5.51e12i)T^{2} \) |
| 71 | \( 1 + (-1.20e6 + 7.74e5i)T + (3.77e12 - 8.27e12i)T^{2} \) |
| 73 | \( 1 + (-5.47e4 + 1.19e5i)T + (-7.23e12 - 8.34e12i)T^{2} \) |
| 79 | \( 1 + (1.72e6 - 1.99e6i)T + (-2.73e12 - 1.90e13i)T^{2} \) |
| 83 | \( 1 + (5.89e6 + 1.73e6i)T + (2.28e13 + 1.46e13i)T^{2} \) |
| 89 | \( 1 + (1.39e6 - 9.71e6i)T + (-4.24e13 - 1.24e13i)T^{2} \) |
| 97 | \( 1 + (-1.11e7 + 3.27e6i)T + (6.79e13 - 4.36e13i)T^{2} \) |
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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
−13.66875863167434406729634523143, −12.77070633562083633761750672293, −10.91892972761632602096643982046, −10.22230340901354114907981197077, −8.655175212451591599220165267701, −7.88319801031733035606418366889, −6.96396303690454184233497574608, −5.76013583169296590604743991938, −3.23508899260256275392419414525, −0.887803567301719858361383707629,
0.086823607871045280941027117674, 2.09835695066615226333609032763, 3.41135621693762472717766399007, 5.37555313178977608627590558383, 7.54322984102768149334509062551, 8.515221097227282763346344899610, 9.789911443788776659643285269861, 10.14246264823977215643208745999, 11.72163225170980108704588273213, 12.22186445978859759680751505808
