L(s) = 1 | + (3.88e5 + 1.54e6i)3-s − 1.99e8i·5-s + 4.03e9·7-s + (−2.23e12 + 1.20e12i)9-s − 4.12e13i·11-s − 9.55e13·13-s + (3.08e14 − 7.74e13i)15-s + 5.48e15i·17-s + 2.57e16·19-s + (1.56e15 + 6.23e15i)21-s + 6.39e17i·23-s + 1.45e18·25-s + (−2.72e18 − 2.99e18i)27-s + 1.27e19i·29-s − 4.74e19·31-s + ⋯ |
L(s) = 1 | + (0.243 + 0.969i)3-s − 0.163i·5-s + 0.0416·7-s + (−0.881 + 0.472i)9-s − 1.19i·11-s − 0.315·13-s + (0.158 − 0.0397i)15-s + 0.553i·17-s + 0.612·19-s + (0.0101 + 0.0403i)21-s + 1.26i·23-s + 0.973·25-s + (−0.673 − 0.739i)27-s + 1.23i·29-s − 1.94·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{27}{2})\) |
\(\approx\) |
\(0.8962426947\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8962426947\) |
\(L(14)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3.88e5 - 1.54e6i)T \) |
good | 5 | \( 1 + 1.99e8iT - 1.49e18T^{2} \) |
| 7 | \( 1 - 4.03e9T + 9.38e21T^{2} \) |
| 11 | \( 1 + 4.12e13iT - 1.19e27T^{2} \) |
| 13 | \( 1 + 9.55e13T + 9.17e28T^{2} \) |
| 17 | \( 1 - 5.48e15iT - 9.81e31T^{2} \) |
| 19 | \( 1 - 2.57e16T + 1.76e33T^{2} \) |
| 23 | \( 1 - 6.39e17iT - 2.54e35T^{2} \) |
| 29 | \( 1 - 1.27e19iT - 1.05e38T^{2} \) |
| 31 | \( 1 + 4.74e19T + 5.96e38T^{2} \) |
| 37 | \( 1 + 6.19e19T + 5.93e40T^{2} \) |
| 41 | \( 1 + 1.36e20iT - 8.55e41T^{2} \) |
| 43 | \( 1 - 2.89e21T + 2.95e42T^{2} \) |
| 47 | \( 1 + 1.44e21iT - 2.98e43T^{2} \) |
| 53 | \( 1 - 1.68e22iT - 6.77e44T^{2} \) |
| 59 | \( 1 + 1.69e23iT - 1.10e46T^{2} \) |
| 61 | \( 1 - 1.07e23T + 2.62e46T^{2} \) |
| 67 | \( 1 + 7.95e22T + 3.00e47T^{2} \) |
| 71 | \( 1 + 1.77e24iT - 1.35e48T^{2} \) |
| 73 | \( 1 + 2.59e24T + 2.79e48T^{2} \) |
| 79 | \( 1 + 5.88e24T + 2.17e49T^{2} \) |
| 83 | \( 1 + 7.37e24iT - 7.87e49T^{2} \) |
| 89 | \( 1 - 7.84e24iT - 4.83e50T^{2} \) |
| 97 | \( 1 + 7.65e25T + 4.52e51T^{2} \) |
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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
−10.65209746930660441120645560872, −9.376581110349050047220587283499, −8.663665649020366574950861785588, −7.44816407477982027273149595660, −5.82052858406183172076794250841, −5.05339511979290584133322909982, −3.71676116832961440884431192344, −3.03487755510861801741763835321, −1.54201182034488602594181273061, −0.16896917294628039530735106349,
0.933043577740483969132706376996, 2.07482919289702022072734448514, 2.86287987496939010550441785634, 4.35366992981099277644200613306, 5.63614860321354875929831072173, 6.93140710287538536857674510603, 7.49691679136827893898521660723, 8.772343980828289267508791109050, 9.846090250573517562244945974226, 11.22424601075550780719267655252