| L(s) = 1 | + (5.73 − 5.73i)2-s − 33.8i·4-s + (92.1 + 92.1i)7-s + (−10.3 − 10.3i)8-s + 126. i·11-s + (−415. + 415. i)13-s + 1.05e3·14-s + 963.·16-s + (−1.14e3 + 1.14e3i)17-s + 710. i·19-s + (723. + 723. i)22-s + (3.00e3 + 3.00e3i)23-s + 4.76e3i·26-s + (3.11e3 − 3.11e3i)28-s − 7.15e3·29-s + ⋯ |
| L(s) = 1 | + (1.01 − 1.01i)2-s − 1.05i·4-s + (0.710 + 0.710i)7-s + (−0.0571 − 0.0571i)8-s + 0.314i·11-s + (−0.681 + 0.681i)13-s + 1.44·14-s + 0.940·16-s + (−0.958 + 0.958i)17-s + 0.451i·19-s + (0.318 + 0.318i)22-s + (1.18 + 1.18i)23-s + 1.38i·26-s + (0.750 − 0.750i)28-s − 1.57·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.527646813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.527646813\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-5.73 + 5.73i)T - 32iT^{2} \) |
| 7 | \( 1 + (-92.1 - 92.1i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 126. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (415. - 415. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.14e3 - 1.14e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 710. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-3.00e3 - 3.00e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 7.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.94e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-6.24e3 - 6.24e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.13e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.30e4 + 1.30e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-125. + 125. i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.60e3 + 1.60e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 1.48e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.25e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.73e4 - 2.73e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 2.97e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (3.13e4 - 3.13e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 6.98e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (9.92e3 + 9.92e3i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 3.30e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.06e4 - 3.06e4i)T + 8.58e9iT^{2} \) |
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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
−11.53188591161883894591995671056, −10.86527127518124601679302436562, −9.663096138858561189273458278551, −8.557988278760647616063745943395, −7.31236856944441324266231794608, −5.78286373168542209775076535334, −4.83802770343721078908106914606, −3.88521564588830622067793357499, −2.44485948203223168334283805036, −1.60718127170969653317206889718,
0.74958822886305140401494020995, 2.82225692833183321768115182574, 4.40684071907339457481674712576, 4.94332650287427319161745422618, 6.20842323338223711554131942061, 7.22666946305807542007493751460, 7.896495707414360027366706553411, 9.256105485941566191026780206422, 10.59691539228479240451959854875, 11.41478454639637990751647453944
