L(s) = 1 | + 41.3i·2-s − 37.6i·3-s − 1.19e3·4-s + (1.13e3 + 810. i)5-s + 1.55e3·6-s + 5.31e3i·7-s − 2.82e4i·8-s + 1.82e4·9-s + (−3.35e4 + 4.70e4i)10-s + 1.04e4·11-s + 4.49e4i·12-s − 7.96e4i·13-s − 2.19e5·14-s + (3.05e4 − 4.28e4i)15-s + 5.54e5·16-s − 3.13e5i·17-s + ⋯ |
L(s) = 1 | + 1.82i·2-s − 0.268i·3-s − 2.33·4-s + (0.814 + 0.580i)5-s + 0.489·6-s + 0.836i·7-s − 2.43i·8-s + 0.928·9-s + (−1.05 + 1.48i)10-s + 0.214·11-s + 0.626i·12-s − 0.773i·13-s − 1.52·14-s + (0.155 − 0.218i)15-s + 2.11·16-s − 0.911i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.401675 + 1.25623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.401675 + 1.25623i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.13e3 - 810. i)T \) |
good | 2 | \( 1 - 41.3iT - 512T^{2} \) |
| 3 | \( 1 + 37.6iT - 1.96e4T^{2} \) |
| 7 | \( 1 - 5.31e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 - 1.04e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.96e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 3.13e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 - 2.46e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 7.21e5iT - 1.80e12T^{2} \) |
| 29 | \( 1 + 2.56e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.29e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.40e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 1.70e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.92e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 4.10e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 5.67e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 1.60e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.33e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.80e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 8.97e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 7.60e7iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 4.10e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.21e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 2.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.03e8iT - 7.60e17T^{2} \) |
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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
−22.88370253057141154788389003688, −21.72889698142492311186561281409, −18.53153659063344673451850935826, −17.76341186864503655243786273213, −15.96632906290911725740572593786, −14.67760821266155244634966781811, −13.18334852706229502296228920541, −9.437150296272990796069135813970, −7.28687580313160429112295426927, −5.63865186807939055305227023641,
1.44920740204565842402946020490, 4.27132325936125202902816454889, 9.368755920511361841287164112360, 10.60380997660035783241402628780, 12.59051006570527268560620301989, 13.83244558470436788780220785247, 16.99921690732488005546092096965, 18.61720263707818649312312787820, 20.15969939062536781994329171645, 21.15813871430524449751723642620