| L(s) = 1 | − 0.898·5-s + 2.82i·7-s − 4.38i·11-s − 13.7·13-s − 17.5·17-s + 4.38i·19-s − 22.0i·23-s − 24.1·25-s + 44.4·29-s + 53.1i·31-s − 2.54i·35-s + 35.1·37-s + 37.5·41-s − 49.6i·43-s − 38.4i·47-s + ⋯ |
| L(s) = 1 | − 0.179·5-s + 0.404i·7-s − 0.398i·11-s − 1.06·13-s − 1.03·17-s + 0.230i·19-s − 0.958i·23-s − 0.967·25-s + 1.53·29-s + 1.71i·31-s − 0.0726i·35-s + 0.951·37-s + 0.916·41-s − 1.15i·43-s − 0.818i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.521343128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.521343128\) |
| \(L(2)\) |
|
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 | \( 1 + 0.898T + 25T^{2} \) |
| 7 | \( 1 - 2.82iT - 49T^{2} \) |
| 11 | \( 1 + 4.38iT - 121T^{2} \) |
| 13 | \( 1 + 13.7T + 169T^{2} \) |
| 17 | \( 1 + 17.5T + 289T^{2} \) |
| 19 | \( 1 - 4.38iT - 361T^{2} \) |
| 23 | \( 1 + 22.0iT - 529T^{2} \) |
| 29 | \( 1 - 44.4T + 841T^{2} \) |
| 31 | \( 1 - 53.1iT - 961T^{2} \) |
| 37 | \( 1 - 35.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 37.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 49.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 38.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 1.70T + 2.80e3T^{2} \) |
| 59 | \( 1 + 34.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 24.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 93.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 123. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 131. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 110. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 73.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 105.T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.675439103587577222619219593658, −8.252705903599588789233427101270, −7.18839127593794776950278101409, −6.58679787185102227384294357739, −5.66107086583649003649566923687, −4.81435311677956656673880705516, −4.05351957875723587519844221798, −2.85764398403672776530846142503, −2.13304077810048205479619027763, −0.60758852781387371657047967876,
0.63287447952298377357710948555, 2.05514014509180295184303632940, 2.89747054395054185065174061937, 4.23957768000489075073527152297, 4.56146552710111278721925219465, 5.75079821356121455830113164756, 6.52006843153475576127804477703, 7.50125138493835270593360408029, 7.78072481408398811146984967162, 8.912667959243972071021132720044
