| L(s) = 1 | + (−8.76 + 5.05i)2-s + (−12.8 + 22.1i)4-s + 328. i·5-s + (−1.23e3 − 715. i)7-s − 1.55e3i·8-s + (−1.66e3 − 2.87e3i)10-s + (1.04e3 − 605. i)11-s + (7.60e3 + 2.22e3i)13-s + 1.44e4·14-s + (6.22e3 + 1.07e4i)16-s + (9.44e3 − 1.63e4i)17-s + (6.06e3 + 3.49e3i)19-s + (−7.27e3 − 4.20e3i)20-s + (−6.13e3 + 1.06e4i)22-s + (−2.96e4 − 5.13e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.774 + 0.447i)2-s + (−0.100 + 0.173i)4-s + 1.17i·5-s + (−1.36 − 0.787i)7-s − 1.07i·8-s + (−0.525 − 0.909i)10-s + (0.237 − 0.137i)11-s + (0.959 + 0.280i)13-s + 1.40·14-s + (0.379 + 0.658i)16-s + (0.466 − 0.807i)17-s + (0.202 + 0.117i)19-s + (−0.203 − 0.117i)20-s + (−0.122 + 0.212i)22-s + (−0.508 − 0.879i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8713640665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8713640665\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (-7.60e3 - 2.22e3i)T \) |
| good | 2 | \( 1 + (8.76 - 5.05i)T + (64 - 110. i)T^{2} \) |
| 5 | \( 1 - 328. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + (1.23e3 + 715. i)T + (4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.04e3 + 605. i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-9.44e3 + 1.63e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-6.06e3 - 3.49e3i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.96e4 + 5.13e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (3.61e4 + 6.26e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 3.01e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + (2.00e5 - 1.15e5i)T + (4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-4.99e5 + 2.88e5i)T + (9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (3.16e5 - 5.49e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 3.76e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 7.41e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (8.21e5 + 4.74e5i)T + (1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (5.61e5 - 9.72e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.33e6 + 1.34e6i)T + (3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-3.65e6 - 2.11e6i)T + (4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 - 5.03e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 8.23e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.63e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + (-8.44e6 + 4.87e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (8.02e6 + 4.63e6i)T + (4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.44469529909621729219264509693, −11.00328784759227514593545356931, −10.05415516349822380067590926292, −9.297477951032480204040532280554, −7.920031180273126948444487562544, −6.85887948509979364785167508406, −6.34113754597292291957854032751, −3.90151043715678725754135462175, −3.05554125382490630839444916605, −0.68529713539360309546124998631,
0.58493437348931810642436850953, 1.77252224622778106418639727552, 3.51408779333849302535798350917, 5.27491759177858379134467848662, 6.16469386722523026679327471735, 8.105584168905433084959992595952, 9.051331713585770479490351011042, 9.546822253378458918515068552601, 10.68885737677679671505750472066, 12.00917297434496140654550463065
