| L(s) = 1 | − 0.742i·2-s + 3.44·4-s + 3.74·5-s + 7.91i·7-s − 5.52i·8-s − 2.78i·10-s + 16.2·11-s − 11.6·13-s + 5.87·14-s + 9.69·16-s + (0.359 + 16.9i)17-s + 0.659·19-s + 12.9·20-s − 12.0i·22-s + 11.8·23-s + ⋯ |
| L(s) = 1 | − 0.371i·2-s + 0.862·4-s + 0.749·5-s + 1.13i·7-s − 0.690i·8-s − 0.278i·10-s + 1.47·11-s − 0.892·13-s + 0.419·14-s + 0.605·16-s + (0.0211 + 0.999i)17-s + 0.0347·19-s + 0.646·20-s − 0.547i·22-s + 0.515·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0211i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.574320120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.574320120\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + (-0.359 - 16.9i)T \) |
| good | 2 | \( 1 + 0.742iT - 4T^{2} \) |
| 5 | \( 1 - 3.74T + 25T^{2} \) |
| 7 | \( 1 - 7.91iT - 49T^{2} \) |
| 11 | \( 1 - 16.2T + 121T^{2} \) |
| 13 | \( 1 + 11.6T + 169T^{2} \) |
| 19 | \( 1 - 0.659T + 361T^{2} \) |
| 23 | \( 1 - 11.8T + 529T^{2} \) |
| 29 | \( 1 + 1.42T + 841T^{2} \) |
| 31 | \( 1 - 9.36iT - 961T^{2} \) |
| 37 | \( 1 - 19.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 32.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 33.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 72.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 11.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 12.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 54.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 77.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 81.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 67.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 95.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 122. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 47.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 27.5iT - 9.40e3T^{2} \) |
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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
−10.92961775104836854948017685191, −9.889044418853017141903425999595, −9.303644966316974431652154990790, −8.251190081688918717473878715871, −6.89112655483339850249162623937, −6.24163138765337349450425118586, −5.31398529658068039546239245134, −3.72217109867227299252740443265, −2.43906949649112978907925307036, −1.57674105865419051225479385829,
1.21855475236895089319840370056, 2.56852293107740459833377119589, 4.00739916033666938060986302489, 5.28092960671232677322589431292, 6.40942018729057611925918966085, 7.04123685938187993834281845859, 7.79287806940517415840258694003, 9.248843302480119177121526505728, 9.868191134841776814533303637216, 10.89808265793969083913365805265
