L(s) = 1 | + 0.843i·2-s + 179. i·3-s + 511.·4-s + (−568. + 1.27e3i)5-s − 151.·6-s − 8.71e3i·7-s + 863. i·8-s − 1.24e4·9-s + (−1.07e3 − 479. i)10-s + 4.45e4·11-s + 9.16e4i·12-s − 2.14e4i·13-s + 7.35e3·14-s + (−2.28e5 − 1.01e5i)15-s + 2.61e5·16-s − 3.00e5i·17-s + ⋯ |
L(s) = 1 | + 0.0372i·2-s + 1.27i·3-s + 0.998·4-s + (−0.406 + 0.913i)5-s − 0.0476·6-s − 1.37i·7-s + 0.0745i·8-s − 0.632·9-s + (−0.0340 − 0.0151i)10-s + 0.917·11-s + 1.27i·12-s − 0.208i·13-s + 0.0511·14-s + (−1.16 − 0.519i)15-s + 0.995·16-s − 0.871i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.25982 + 0.818238i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25982 + 0.818238i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (568. - 1.27e3i)T \) |
good | 2 | \( 1 - 0.843iT - 512T^{2} \) |
| 3 | \( 1 - 179. iT - 1.96e4T^{2} \) |
| 7 | \( 1 + 8.71e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 - 4.45e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.14e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 3.00e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + 5.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.50e5iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 8.03e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.99e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.53e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 2.54e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.32e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 3.77e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 4.79e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 7.00e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.26e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.66e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 6.59e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.47e7iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 4.66e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.01e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 5.54e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.39e8iT - 7.60e17T^{2} \) |
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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
−22.13157700713439453435668565862, −20.70070861482831749089040171645, −19.59015321128470958522070642103, −16.98643246344756846455524648869, −15.73086688659783430906865798029, −14.46261027238274674268282166665, −11.27038605898789196311788413999, −10.21196647938936155377652079279, −7.05083715819696954166285473446, −3.73921493109274457842441622083,
1.76511031061647646631543012965, 6.37816424446454197007858084399, 8.413826415017186276113723017656, 11.84098952926041106032905428276, 12.64232969756892670754402226057, 15.17597332297266355341544155098, 16.88656601889865950722610501348, 18.83088772989177603272164560100, 19.83773674695818149447442954709, 21.52816635080805168348419066953