L(s) = 1 | − 2i·5-s + 9.79i·7-s − 19.5·11-s + 21·25-s − 50i·29-s − 48.9i·31-s + 19.5·35-s − 46.9·49-s − 94i·53-s + 39.1i·55-s + 117.·59-s + 50·73-s − 191. i·77-s + 146. i·79-s − 97.9·83-s + ⋯ |
L(s) = 1 | − 0.400i·5-s + 1.39i·7-s − 1.78·11-s + 0.839·25-s − 1.72i·29-s − 1.58i·31-s + 0.559·35-s − 0.959·49-s − 1.77i·53-s + 0.712i·55-s + 1.99·59-s + 0.684·73-s − 2.49i·77-s + 1.86i·79-s − 1.18·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9715480761\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9715480761\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2iT - 25T^{2} \) |
| 7 | \( 1 - 9.79iT - 49T^{2} \) |
| 11 | \( 1 + 19.5T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 50iT - 841T^{2} \) |
| 31 | \( 1 + 48.9iT - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 94iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 117.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 50T + 5.32e3T^{2} \) |
| 79 | \( 1 - 146. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 97.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 + 190T + 9.40e3T^{2} \) |
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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
−9.484839687406892852801183362862, −8.277572098925395953914952393266, −8.185090234290063424400320677495, −6.89960105869955702143506270469, −5.67290001916986564908025719962, −5.38668005720655873395109338967, −4.27265743869430506933979926611, −2.79078265767023872827905244865, −2.18684730535767663553017238104, −0.31453190218697586412276164792,
1.12355760716255419722170827842, 2.68470493628005419198723418462, 3.53043892537416800075037921730, 4.71481015480713285514728213170, 5.41407913558596952065603177629, 6.76512107375981661011391571533, 7.25538197931990242484647531083, 8.037676968147230794508588794071, 8.939748326913749930361121673862, 10.15446460695774725127509977989