L(s) = 1 | + (1.71e3 + 1.35e3i)3-s + 9.73e4i·5-s − 5.26e5·7-s + (1.09e6 + 4.65e6i)9-s − 1.93e7i·11-s − 3.32e7·13-s + (−1.32e8 + 1.66e8i)15-s + 5.75e8i·17-s − 1.45e9·19-s + (−9.02e8 − 7.14e8i)21-s + 4.29e9i·23-s − 3.36e9·25-s + (−4.44e9 + 9.46e9i)27-s − 1.71e10i·29-s + 3.40e10·31-s + ⋯ |
L(s) = 1 | + (0.783 + 0.620i)3-s + 1.24i·5-s − 0.639·7-s + (0.228 + 0.973i)9-s − 0.993i·11-s − 0.530·13-s + (−0.773 + 0.976i)15-s + 1.40i·17-s − 1.62·19-s + (−0.500 − 0.396i)21-s + 1.26i·23-s − 0.551·25-s + (−0.425 + 0.905i)27-s − 0.995i·29-s + 1.23·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.620i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.783 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(0.546755 + 1.57071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.546755 + 1.57071i\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.71e3 - 1.35e3i)T \) |
good | 5 | \( 1 - 9.73e4iT - 6.10e9T^{2} \) |
| 7 | \( 1 + 5.26e5T + 6.78e11T^{2} \) |
| 11 | \( 1 + 1.93e7iT - 3.79e14T^{2} \) |
| 13 | \( 1 + 3.32e7T + 3.93e15T^{2} \) |
| 17 | \( 1 - 5.75e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 1.45e9T + 7.99e17T^{2} \) |
| 23 | \( 1 - 4.29e9iT - 1.15e19T^{2} \) |
| 29 | \( 1 + 1.71e10iT - 2.97e20T^{2} \) |
| 31 | \( 1 - 3.40e10T + 7.56e20T^{2} \) |
| 37 | \( 1 - 8.31e10T + 9.01e21T^{2} \) |
| 41 | \( 1 - 2.11e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 1.48e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + 4.40e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 - 1.70e12iT - 1.37e24T^{2} \) |
| 59 | \( 1 + 1.79e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 5.48e12T + 9.87e24T^{2} \) |
| 67 | \( 1 - 1.01e13T + 3.67e25T^{2} \) |
| 71 | \( 1 + 1.61e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 + 8.08e12T + 1.22e26T^{2} \) |
| 79 | \( 1 - 1.32e13T + 3.68e26T^{2} \) |
| 83 | \( 1 - 8.28e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 5.62e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 1.34e12T + 6.52e27T^{2} \) |
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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
−17.05906905319087775967794921593, −15.48495554683527233049002470225, −14.58464009222656308802934068409, −13.25327148780974804921590455035, −10.98645025847358723175777641962, −9.887451366340574418945944882436, −8.160110574231045732073828342687, −6.31956678780733494886625692395, −3.80573945038187321695906676205, −2.54569530200173737761816667678,
0.59052087269575093160410042875, 2.37539328154355663317997818718, 4.55672099594406924624946629760, 6.85508833436717884479189791366, 8.511662838536445449533398278643, 9.678053229493730828684481089117, 12.33986700207610982474666801523, 12.98165708748667407510860877144, 14.58244799752907580769727372132, 16.10718296923772113600009098994