L(s) = 1 | + (0.654 − 1.25i)2-s + i·3-s + (−1.14 − 1.64i)4-s + (−0.206 + 2.22i)5-s + (1.25 + 0.654i)6-s + (0.859 + 2.50i)7-s + (−2.80 + 0.359i)8-s − 9-s + (2.65 + 1.71i)10-s + 4.14i·11-s + (1.64 − 1.14i)12-s − 3.35·13-s + (3.69 + 0.559i)14-s + (−2.22 − 0.206i)15-s + (−1.38 + 3.75i)16-s − 3.18·17-s + ⋯ |
L(s) = 1 | + (0.462 − 0.886i)2-s + 0.577i·3-s + (−0.571 − 0.820i)4-s + (−0.0925 + 0.995i)5-s + (0.511 + 0.267i)6-s + (0.324 + 0.945i)7-s + (−0.991 + 0.127i)8-s − 0.333·9-s + (0.839 + 0.542i)10-s + 1.24i·11-s + (0.473 − 0.330i)12-s − 0.930·13-s + (0.988 + 0.149i)14-s + (−0.574 − 0.0534i)15-s + (−0.346 + 0.938i)16-s − 0.773·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29215 + 0.582397i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29215 + 0.582397i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 + 1.25i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.206 - 2.22i)T \) |
| 7 | \( 1 + (-0.859 - 2.50i)T \) |
good | 11 | \( 1 - 4.14iT - 11T^{2} \) |
| 13 | \( 1 + 3.35T + 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 - 6.52T + 19T^{2} \) |
| 23 | \( 1 - 1.42T + 23T^{2} \) |
| 29 | \( 1 - 8.88T + 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 + 9.54iT - 37T^{2} \) |
| 41 | \( 1 - 3.55iT - 41T^{2} \) |
| 43 | \( 1 - 0.667T + 43T^{2} \) |
| 47 | \( 1 - 0.693iT - 47T^{2} \) |
| 53 | \( 1 - 4.18iT - 53T^{2} \) |
| 59 | \( 1 + 5.96T + 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 7.05T + 67T^{2} \) |
| 71 | \( 1 - 9.50iT - 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 5.25iT - 83T^{2} \) |
| 89 | \( 1 + 2.83iT - 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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
−11.44270679584842554080331396432, −10.42467103433779438768330661550, −9.763148310102240175844297796119, −9.035023878871589211077463464455, −7.61370116935143805991631724758, −6.41407063232632836755106043568, −5.20326634662597034063499944519, −4.45429491525098164731914264715, −3.02646704542084173048657513385, −2.22435035149540777115697325870,
0.811552717870372228166425167036, 3.12837293457054140289665574955, 4.50398480178542624790113319256, 5.24769111366999075163785338201, 6.41479496620451371028326766080, 7.37107670771942314552412167944, 8.153463040221278019226592528641, 8.844705576265740716372019062753, 9.999874944179312610974270272481, 11.48181208565956576048499126595