| L(s) = 1 | − 0.258i·2-s + 7.93·4-s − 14.5i·7-s − 4.12i·8-s + 49.2·11-s + 72.1i·13-s − 3.75·14-s + 62.3·16-s + 118. i·17-s − 123.·19-s − 12.7i·22-s + 91.4i·23-s + 18.6·26-s − 115. i·28-s + 174.·29-s + ⋯ |
| L(s) = 1 | − 0.0914i·2-s + 0.991·4-s − 0.783i·7-s − 0.182i·8-s + 1.35·11-s + 1.53i·13-s − 0.0716·14-s + 0.974·16-s + 1.68i·17-s − 1.48·19-s − 0.123i·22-s + 0.829i·23-s + 0.140·26-s − 0.777i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.775819968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.775819968\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.258iT - 8T^{2} \) |
| 7 | \( 1 + 14.5iT - 343T^{2} \) |
| 11 | \( 1 - 49.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 118. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 91.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 46.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 364.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 125. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 221. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 13.6iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 239.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 76.0iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 728.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 501. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 397.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.36e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 335. iT - 9.12e5T^{2} \) |
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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
−10.39449246800823800498689538624, −9.322574327266213831475954344028, −8.450823095107822087322875618020, −7.33264386419012854822471944413, −6.56242698272141716308875421911, −6.08458722929550626111649704771, −4.24728976343177361405627388235, −3.78286239579947985653421401020, −2.10348371509837615737129583890, −1.28575157749916072699506230107,
0.828149995391711281771873785559, 2.33022616579725847727764820506, 3.09548739059331655859451273394, 4.56342519994204696079251652982, 5.75684666730261384845037905508, 6.43261027455390484333895962932, 7.29758048640475857949668644319, 8.322191091759754522241541587699, 9.075860399018865763306617892171, 10.16294807774722505265696328435
