| L(s) = 1 | + (3.79 − 3.79i)2-s + 227. i·4-s + (563. − 269. i)5-s + (−1.26e3 + 1.26e3i)7-s + (1.83e3 + 1.83e3i)8-s + (1.11e3 − 3.16e3i)10-s − 2.39e3·11-s + (4.60e3 + 4.60e3i)13-s + 9.59e3i·14-s − 4.42e4·16-s + (−6.29e4 + 6.29e4i)17-s + 1.14e5i·19-s + (6.12e4 + 1.28e5i)20-s + (−9.08e3 + 9.08e3i)22-s + (2.64e5 + 2.64e5i)23-s + ⋯ |
| L(s) = 1 | + (0.237 − 0.237i)2-s + 0.887i·4-s + (0.901 − 0.431i)5-s + (−0.526 + 0.526i)7-s + (0.448 + 0.448i)8-s + (0.111 − 0.316i)10-s − 0.163·11-s + (0.161 + 0.161i)13-s + 0.249i·14-s − 0.674·16-s + (−0.754 + 0.754i)17-s + 0.877i·19-s + (0.383 + 0.800i)20-s + (−0.0388 + 0.0388i)22-s + (0.944 + 0.944i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.50320 + 1.35020i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50320 + 1.35020i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-563. + 269. i)T \) |
| good | 2 | \( 1 + (-3.79 + 3.79i)T - 256iT^{2} \) |
| 7 | \( 1 + (1.26e3 - 1.26e3i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + 2.39e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + (-4.60e3 - 4.60e3i)T + 8.15e8iT^{2} \) |
| 17 | \( 1 + (6.29e4 - 6.29e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 - 1.14e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-2.64e5 - 2.64e5i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 - 1.26e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 4.62e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (7.85e5 - 7.85e5i)T - 3.51e12iT^{2} \) |
| 41 | \( 1 - 1.91e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (3.06e6 + 3.06e6i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 + (-1.27e6 + 1.27e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 + (5.22e6 + 5.22e6i)T + 6.22e13iT^{2} \) |
| 59 | \( 1 + 2.36e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.07e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + (-2.21e7 + 2.21e7i)T - 4.06e14iT^{2} \) |
| 71 | \( 1 - 2.73e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-4.75e6 - 4.75e6i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 - 8.87e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-2.53e7 - 2.53e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 - 6.89e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-5.46e7 + 5.46e7i)T - 7.83e15iT^{2} \) |
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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
−14.03601715680990205302313853372, −12.99258941197859649252390019919, −12.36689108903476381477289112331, −10.88432025221444737800806171199, −9.375874326633424678360273476642, −8.359229005888478018608299501064, −6.62143354908484608192465578586, −5.11654811745532087537668287380, −3.37249016045644819775674965762, −1.86054360857271800029772313658,
0.69062456842212257931386364014, 2.53483430738691313987224555922, 4.69410282519487441027613325160, 6.12784499046631994412046247084, 7.02456681383139201292073231786, 9.207731442675819447142725686919, 10.18359380671941559044303168159, 11.13591859480426814385198349285, 13.19041141239311184664654410174, 13.73530157785410391983335633799
