| L(s) = 1 | − 2.55·2-s + 4.52·4-s − i·5-s − 0.112i·7-s − 6.43·8-s + 2.55i·10-s + (2.27 − 2.41i)11-s − 2.16i·13-s + 0.286i·14-s + 7.39·16-s − 3.32·17-s − 2.43i·19-s − 4.52i·20-s + (−5.80 + 6.16i)22-s + 4.21i·23-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 2.26·4-s − 0.447i·5-s − 0.0424i·7-s − 2.27·8-s + 0.807i·10-s + (0.685 − 0.727i)11-s − 0.599i·13-s + 0.0765i·14-s + 1.84·16-s − 0.807·17-s − 0.558i·19-s − 1.01i·20-s + (−1.23 + 1.31i)22-s + 0.878i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.326033 - 0.375216i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.326033 - 0.375216i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 + (-2.27 + 2.41i)T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 7 | \( 1 + 0.112iT - 7T^{2} \) |
| 13 | \( 1 + 2.16iT - 13T^{2} \) |
| 17 | \( 1 + 3.32T + 17T^{2} \) |
| 19 | \( 1 + 2.43iT - 19T^{2} \) |
| 23 | \( 1 - 4.21iT - 23T^{2} \) |
| 29 | \( 1 - 9.11T + 29T^{2} \) |
| 31 | \( 1 + 6.54T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 + 8.32T + 41T^{2} \) |
| 43 | \( 1 + 6.72iT - 43T^{2} \) |
| 47 | \( 1 + 12.7iT - 47T^{2} \) |
| 53 | \( 1 + 10.6iT - 53T^{2} \) |
| 59 | \( 1 + 4.79iT - 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 1.08T + 67T^{2} \) |
| 71 | \( 1 - 11.6iT - 71T^{2} \) |
| 73 | \( 1 - 3.88iT - 73T^{2} \) |
| 79 | \( 1 + 0.241iT - 79T^{2} \) |
| 83 | \( 1 - 0.512T + 83T^{2} \) |
| 89 | \( 1 + 13.9iT - 89T^{2} \) |
| 97 | \( 1 - 1.91T + 97T^{2} \) |
| 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
−10.47606172907543487713878879150, −9.717887494147501733328959156258, −8.656665866557015274532744978155, −8.529808684496200928770125987947, −7.23583987409954296593639166037, −6.52816683753388006268864721252, −5.26444354499295075882825384222, −3.47671618669951102714614672895, −1.95249346081963448197903823610, −0.55021033797586653252902634576,
1.51251136378697494054721235373, 2.68344904369815056435640374117, 4.37610481764982821164833932613, 6.24557180185856348293762321596, 6.85466985597282406230703318101, 7.68728992453046439684532312151, 8.739601354910739225968998532550, 9.282849499435520598260623291421, 10.25400313632501446236123270152, 10.78820427128385574383338406967
