L(s) = 1 | + 5.52·2-s + 15.5i·3-s − 33.4·4-s − 66.9i·5-s + 86.1i·6-s − 538.·8-s − 243·9-s − 370. i·10-s + 1.72e3·11-s − 521. i·12-s − 2.80e3i·13-s + 1.04e3·15-s − 836.·16-s + 6.14e3i·17-s − 1.34e3·18-s + 8.93e3i·19-s + ⋯ |
L(s) = 1 | + 0.690·2-s + 0.577i·3-s − 0.522·4-s − 0.535i·5-s + 0.398i·6-s − 1.05·8-s − 0.333·9-s − 0.370i·10-s + 1.29·11-s − 0.301i·12-s − 1.27i·13-s + 0.309·15-s − 0.204·16-s + 1.25i·17-s − 0.230·18-s + 1.30i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.279421643\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.279421643\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 15.5iT \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 5.52T + 64T^{2} \) |
| 5 | \( 1 + 66.9iT - 1.56e4T^{2} \) |
| 11 | \( 1 - 1.72e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 2.80e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.14e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 8.93e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 9.90e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 1.36e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 2.52e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 2.27e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.78e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 7.36e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.39e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.65e4T + 2.21e10T^{2} \) |
| 59 | \( 1 - 7.15e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.63e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 5.48e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.65e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 3.33e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 4.10e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 7.19e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.63e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 6.51e5iT - 8.32e11T^{2} \) |
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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
−12.54336592190719072952791355766, −11.11979289514418207047127801261, −9.945869378320283691978486186347, −8.987410158171263680327308078748, −8.144255344538956302426679961865, −6.23358266558794925297047354359, −5.30631344488908243697894922401, −4.18175963159195489537073850209, −3.31408060215772836685577922180, −1.06402532321720691182323759061,
0.71048645928090666030012604580, 2.53135947836533200955198029359, 3.88633271593538513934564387767, 5.04032353947480114089377924706, 6.48695713739567690187709946364, 7.13691391060641285250202103218, 8.932163967214271594128709777311, 9.395788631681376380908224976225, 11.26883467894353173234445956218, 11.81030684484138409299187211339