| L(s) = 1 | − 6.57e3i·3-s + 30·5-s + 3.03e6i·7-s − 1.96e5·9-s − 3.51e8i·11-s + 3.70e7·13-s − 1.97e5i·15-s − 1.38e9·17-s + 1.88e10i·19-s + 1.99e10·21-s + 3.55e10i·23-s − 1.52e11·25-s − 2.81e11i·27-s + 7.22e11·29-s − 9.61e11i·31-s + ⋯ |
| L(s) = 1 | − 1.00i·3-s + 7.67e−5·5-s + 0.527i·7-s − 0.00456·9-s − 1.63i·11-s + 0.0454·13-s − 7.69e − 5i·15-s − 0.198·17-s + 1.10i·19-s + 0.528·21-s + 0.453i·23-s − 0.999·25-s − 0.997i·27-s + 1.44·29-s − 1.12i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(1.230150304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230150304\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 6.57e3iT - 4.30e7T^{2} \) |
| 5 | \( 1 - 30T + 1.52e11T^{2} \) |
| 7 | \( 1 - 3.03e6iT - 3.32e13T^{2} \) |
| 11 | \( 1 + 3.51e8iT - 4.59e16T^{2} \) |
| 13 | \( 1 - 3.70e7T + 6.65e17T^{2} \) |
| 17 | \( 1 + 1.38e9T + 4.86e19T^{2} \) |
| 19 | \( 1 - 1.88e10iT - 2.88e20T^{2} \) |
| 23 | \( 1 - 3.55e10iT - 6.13e21T^{2} \) |
| 29 | \( 1 - 7.22e11T + 2.50e23T^{2} \) |
| 31 | \( 1 + 9.61e11iT - 7.27e23T^{2} \) |
| 37 | \( 1 - 5.01e12T + 1.23e25T^{2} \) |
| 41 | \( 1 + 6.62e12T + 6.37e25T^{2} \) |
| 43 | \( 1 + 1.67e13iT - 1.36e26T^{2} \) |
| 47 | \( 1 + 3.13e13iT - 5.66e26T^{2} \) |
| 53 | \( 1 + 7.42e13T + 3.87e27T^{2} \) |
| 59 | \( 1 + 1.47e14iT - 2.15e28T^{2} \) |
| 61 | \( 1 + 1.50e14T + 3.67e28T^{2} \) |
| 67 | \( 1 - 3.70e14iT - 1.64e29T^{2} \) |
| 71 | \( 1 - 1.89e14iT - 4.16e29T^{2} \) |
| 73 | \( 1 + 9.00e14T + 6.50e29T^{2} \) |
| 79 | \( 1 - 2.47e15iT - 2.30e30T^{2} \) |
| 83 | \( 1 + 1.19e15iT - 5.07e30T^{2} \) |
| 89 | \( 1 + 1.53e14T + 1.54e31T^{2} \) |
| 97 | \( 1 - 3.92e15T + 6.14e31T^{2} \) |
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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
−11.41096191109919856650437122450, −10.00823950864426747556807256137, −8.570861803626952260173385073672, −7.79012167561866721348656781013, −6.40954270170241407119674159749, −5.64065767703853794625464255586, −3.83908980752354122407763819874, −2.49888259512546111697524156633, −1.33887358175120958240333269375, −0.26928233043721334331995182278,
1.35690509683542542801462726221, 2.83735266517604882177757416966, 4.36461014262743872829330254989, 4.73274113667368332884142027774, 6.56415781206649890310432709107, 7.65209805104696420648584558977, 9.209392741620010644717459702981, 10.00389913042271198570407236063, 10.83960240201976071934596721359, 12.18519142015751338279635259415
