L(s) = 1 | − 5.52·2-s − 33.4·4-s + 66.9i·5-s + 538.·8-s − 370. i·10-s − 1.72e3·11-s − 2.80e3i·13-s − 836.·16-s − 6.14e3i·17-s + 8.93e3i·19-s − 2.24e3i·20-s + 9.53e3·22-s − 9.90e3·23-s + 1.11e4·25-s + 1.55e4i·26-s + ⋯ |
L(s) = 1 | − 0.690·2-s − 0.522·4-s + 0.535i·5-s + 1.05·8-s − 0.370i·10-s − 1.29·11-s − 1.27i·13-s − 0.204·16-s − 1.25i·17-s + 1.30i·19-s − 0.280i·20-s + 0.895·22-s − 0.813·23-s + 0.712·25-s + 0.882i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.2501384725\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2501384725\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 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
−10.28870531225850647549522800382, −9.803861337783901162448935992209, −8.522952644149935630873294313783, −7.913917969274699670415356786626, −7.13036281821090326283140953749, −5.66135791702873662506763551042, −4.90175715715742628419258325473, −3.49598085262665379735333673294, −2.42266389989539049553370128369, −0.852885890635526741297725348350,
0.098251772568345648408968453292, 1.24028169590796622065971850454, 2.47550019288027967251432069286, 4.18957584990310788497538438967, 4.79954621459457778812657645849, 6.01606326240294202404619462883, 7.30009253981536906897842352054, 8.188934449913571942986164438888, 8.858800344845030099564916299063, 9.648106740153551553895772508940