| L(s) = 1 | − 20.2i·2-s − 30.5i·3-s + 103.·4-s − 616.·6-s − 4.01e3i·7-s − 1.24e4i·8-s + 1.87e4·9-s − 4.21e4·11-s − 3.16e3i·12-s − 1.23e5i·13-s − 8.10e4·14-s − 1.98e5·16-s + 3.19e5i·17-s − 3.78e5i·18-s − 1.08e6·19-s + ⋯ |
| L(s) = 1 | − 0.892i·2-s − 0.217i·3-s + 0.202·4-s − 0.194·6-s − 0.631i·7-s − 1.07i·8-s + 0.952·9-s − 0.867·11-s − 0.0441i·12-s − 1.20i·13-s − 0.563·14-s − 0.755·16-s + 0.929i·17-s − 0.850i·18-s − 1.91·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.408940 - 1.73230i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.408940 - 1.73230i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 20.2iT - 512T^{2} \) |
| 3 | \( 1 + 30.5iT - 1.96e4T^{2} \) |
| 7 | \( 1 + 4.01e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 + 4.21e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.23e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 3.19e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + 1.08e6T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.50e6iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 2.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.51e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 2.95e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.42e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 1.35e6iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 9.73e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 7.48e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.11e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.94e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 1.56e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.82e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 5.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.48e6iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 5.99e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.25e8iT - 7.60e17T^{2} \) |
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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
−15.02322953797697355504656428003, −13.02500796737904107199955419111, −12.62068831738398722532902702825, −10.68167755981573362314524521567, −10.28012202099140883907588128922, −8.021727425202398088449442354075, −6.52221893871750887945163231796, −4.17010621746123734304359606362, −2.39394081384911155270807133389, −0.74552763452892293371668596158,
2.21909790519304437450975363184, 4.71079106191059034304430468232, 6.30868199971019862978104645118, 7.58665471166831721862882613091, 9.102952146606983284150116524745, 10.77311692884424013402759595250, 12.19831581826135189399060445489, 13.77612671518556833725221870673, 15.21783681895914817167649892812, 15.81390921829063582372930662984
