L(s) = 1 | − i·3-s − i·5-s − 0.143·7-s − 9-s − 4.80·11-s + 4.78·13-s − 15-s + 4.86i·17-s − 0.379·19-s + 0.143i·21-s + (−1.15 − 4.65i)23-s − 25-s + i·27-s + 8.39·29-s + 2.55i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447i·5-s − 0.0540·7-s − 0.333·9-s − 1.44·11-s + 1.32·13-s − 0.258·15-s + 1.18i·17-s − 0.0870·19-s + 0.0312i·21-s + (−0.240 − 0.970i)23-s − 0.200·25-s + 0.192i·27-s + 1.55·29-s + 0.459i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4659240266\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4659240266\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (1.15 + 4.65i)T \) |
good | 7 | \( 1 + 0.143T + 7T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 - 4.86iT - 17T^{2} \) |
| 19 | \( 1 + 0.379T + 19T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 - 2.55iT - 31T^{2} \) |
| 37 | \( 1 + 5.56iT - 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 + 8.93iT - 47T^{2} \) |
| 53 | \( 1 + 13.3iT - 53T^{2} \) |
| 59 | \( 1 + 6.57iT - 59T^{2} \) |
| 61 | \( 1 - 7.49iT - 61T^{2} \) |
| 67 | \( 1 + 4.96T + 67T^{2} \) |
| 71 | \( 1 - 6.77iT - 71T^{2} \) |
| 73 | \( 1 + 8.04T + 73T^{2} \) |
| 79 | \( 1 - 1.53T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 12.6iT - 89T^{2} \) |
| 97 | \( 1 - 5.67iT - 97T^{2} \) |
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
−8.037559921682806942298487139650, −6.96950693975560013787041712692, −6.36232436091260579143998070848, −5.65638943004054095642608487103, −4.96527327731074131053114937247, −4.03599974312019794359448692137, −3.16181259842293855383213572200, −2.21704362487938861276024510945, −1.31006044925885157003085911853, −0.12298720419656661919709730724,
1.36317052535055346994769383428, 2.83376017753052836603458316320, 3.03488585674059580277678824805, 4.18508220902480832198730982794, 4.89624147400049743353277547933, 5.65704297080413163146432244630, 6.29815893617613571547647464115, 7.15421266921786149222102516778, 7.961424158704750592222205159116, 8.406785789005999339832498824248