| L(s) = 1 | + (15.9 − 0.0851i)2-s + (−103. − 42.6i)3-s + (255. − 2.72i)4-s + (−547. + 226. i)5-s + (−1.65e3 − 674. i)6-s + (2.25e3 + 2.25e3i)7-s + (4.09e3 − 65.3i)8-s + (4.15e3 + 4.15e3i)9-s + (−8.73e3 + 3.67e3i)10-s + (1.03e4 − 4.27e3i)11-s + (−2.64e4 − 1.06e4i)12-s + (3.47e4 + 1.44e4i)13-s + (3.62e4 + 3.58e4i)14-s + 6.60e4·15-s + (6.55e4 − 1.39e3i)16-s + 5.62e4i·17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.00531i)2-s + (−1.27 − 0.526i)3-s + (0.999 − 0.0106i)4-s + (−0.875 + 0.362i)5-s + (−1.27 − 0.520i)6-s + (0.939 + 0.939i)7-s + (0.999 − 0.0159i)8-s + (0.633 + 0.633i)9-s + (−0.873 + 0.367i)10-s + (0.704 − 0.291i)11-s + (−1.27 − 0.513i)12-s + (1.21 + 0.504i)13-s + (0.944 + 0.934i)14-s + 1.30·15-s + (0.999 − 0.0212i)16-s + 0.673i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.08438 + 0.662687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08438 + 0.662687i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.9 + 0.0851i)T \) |
| good | 3 | \( 1 + (103. + 42.6i)T + (4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (547. - 226. i)T + (2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-2.25e3 - 2.25e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + (-1.03e4 + 4.27e3i)T + (1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (-3.47e4 - 1.44e4i)T + (5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 - 5.62e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (1.38e4 - 3.33e4i)T + (-1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (2.48e5 - 2.48e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (-4.43e5 + 1.07e6i)T + (-3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 - 1.50e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-2.06e6 + 8.56e5i)T + (2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (2.20e6 + 2.20e6i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (1.43e6 - 5.94e5i)T + (8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 - 3.09e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (6.57e5 + 1.58e6i)T + (-4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (1.24e6 + 3.01e6i)T + (-1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (-1.01e6 + 2.45e6i)T + (-1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (2.69e7 + 1.11e7i)T + (2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (-1.70e7 - 1.70e7i)T + 6.45e14iT^{2} \) |
| 73 | \( 1 + (-3.36e7 - 3.36e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 - 3.93e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (2.55e7 - 6.15e7i)T + (-1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (-6.66e7 + 6.66e7i)T - 3.93e15iT^{2} \) |
| 97 | \( 1 + 3.45e7T + 7.83e15T^{2} \) |
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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
−15.15503310143357450048662758468, −13.86958569301683440533221047129, −12.23350763092010692253016939154, −11.64317626522269062566326638737, −10.98853566303961945763954037828, −8.145521592653612492662094531782, −6.55006073633129949423511216118, −5.59433730394246059145432079778, −3.93369239226884025680073132796, −1.53449177168482266873629184212,
0.907333010935768853085778227637, 4.02848290837078901261662440560, 4.80312236556074905992759749550, 6.35270307981019654995254995992, 7.926675416860093411244185503210, 10.52830425713100125127991697022, 11.34884875701143027803901351954, 12.10264303468603881808464684294, 13.65350233901654619304590975463, 14.99198725397571201541749285681
