| L(s) = 1 | + 10.1i·2-s + 23.0i·3-s − 71.4·4-s − 234.·6-s + 51.0i·7-s − 401. i·8-s − 288.·9-s − 512.·11-s − 1.64e3i·12-s + 169i·13-s − 519.·14-s + 1.79e3·16-s − 978. i·17-s − 2.93e3i·18-s − 1.35e3·19-s + ⋯ |
| L(s) = 1 | + 1.79i·2-s + 1.47i·3-s − 2.23·4-s − 2.65·6-s + 0.393i·7-s − 2.21i·8-s − 1.18·9-s − 1.27·11-s − 3.30i·12-s + 0.277i·13-s − 0.708·14-s + 1.75·16-s − 0.821i·17-s − 2.13i·18-s − 0.860·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.03565927271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03565927271\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 2 | \( 1 - 10.1iT - 32T^{2} \) |
| 3 | \( 1 - 23.0iT - 243T^{2} \) |
| 7 | \( 1 - 51.0iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 512.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 978. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.35e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 11.3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 8.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 759.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.27e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.81e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.95e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.20e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.38e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.81e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.62e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.76e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 7.57e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.66e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.68e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.23e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.75e4iT - 8.58e9T^{2} \) |
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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
−10.41867012072670629244876013984, −9.849864656474009795496182155992, −8.766150450830092125371639424793, −8.260437818607678951065681159241, −7.00771610518488531955144629018, −5.94136962740314497622359387354, −4.92543621414545199296707327040, −4.55466114018571159554847456152, −3.00707448815912806399961138979, −0.01155801536580284930525462507,
0.929598601928396284561748694266, 2.01807941354538640865905245492, 2.79419867308098144629615891896, 4.14766875618198819825954020424, 5.49255324502514397848133366215, 6.86694743870784475257741364114, 8.056340022975117447383181478706, 8.679601692738038288195090799201, 10.25287629847047053555629505772, 10.57979050213745466927738670276
