L(s) = 1 | + 81i·3-s + 553. i·5-s + 1.24e4·7-s − 6.56e3·9-s − 3.97e4i·11-s − 9.37e4i·13-s − 4.48e4·15-s − 2.11e5·17-s + 2.01e5i·19-s + 1.01e6i·21-s − 1.90e6·23-s + 1.64e6·25-s − 5.31e5i·27-s + 4.59e6i·29-s + 1.17e6·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.396i·5-s + 1.96·7-s − 0.333·9-s − 0.818i·11-s − 0.910i·13-s − 0.228·15-s − 0.613·17-s + 0.354i·19-s + 1.13i·21-s − 1.41·23-s + 0.842·25-s − 0.192i·27-s + 1.20i·29-s + 0.227·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(2.950426562\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.950426562\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 81iT \) |
good | 5 | \( 1 - 553. iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 1.24e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.97e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + 9.37e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 2.11e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.01e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + 1.90e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.59e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 - 1.17e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.71e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 7.69e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.79e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 1.24e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.48e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 4.27e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 + 3.97e6iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 7.11e6iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 1.75e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.15e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.06e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.10e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 7.60e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.20e9T + 7.60e17T^{2} \) |
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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
−10.25293754873476181328994268852, −8.722611182074966840447738073557, −8.301213093253925247855232764696, −7.32883189621571338676398882814, −5.93066363109770926217710396581, −5.11602237382361518259048770062, −4.23948982121395960478471228462, −3.09019310334355565201452725120, −1.92671301421582370984473478778, −0.799924679040807008262850692497,
0.67920916625737333614039025373, 1.79053880031731331956095415423, 2.22580114212344125859384132009, 4.27183432480192449337044401890, 4.72025295271740783153991130035, 5.90756612331910111485792925577, 7.09429947132912327340714029686, 7.892994650760235525625545690066, 8.587724070065957181850515931070, 9.558547029985468986725702609063