L(s) = 1 | − 1.38·5-s − 7.59i·7-s + 7.52i·11-s + 14.7·13-s + 30.7·17-s − 4.35i·19-s − 4.11i·23-s − 23.0·25-s + 18.9·29-s − 3.84i·31-s + 10.5i·35-s − 42.0·37-s + 49.6·41-s + 5.08i·43-s + 32.6i·47-s + ⋯ |
L(s) = 1 | − 0.276·5-s − 1.08i·7-s + 0.684i·11-s + 1.13·13-s + 1.80·17-s − 0.229i·19-s − 0.179i·23-s − 0.923·25-s + 0.653·29-s − 0.123i·31-s + 0.300i·35-s − 1.13·37-s + 1.21·41-s + 0.118i·43-s + 0.694i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.203105802\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.203105802\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 5 | \( 1 + 1.38T + 25T^{2} \) |
| 7 | \( 1 + 7.59iT - 49T^{2} \) |
| 11 | \( 1 - 7.52iT - 121T^{2} \) |
| 13 | \( 1 - 14.7T + 169T^{2} \) |
| 17 | \( 1 - 30.7T + 289T^{2} \) |
| 23 | \( 1 + 4.11iT - 529T^{2} \) |
| 29 | \( 1 - 18.9T + 841T^{2} \) |
| 31 | \( 1 + 3.84iT - 961T^{2} \) |
| 37 | \( 1 + 42.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 49.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 5.08iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 32.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 75.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 75.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 50.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 45.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 115.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 67.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 22.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 95.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 43.1T + 9.40e3T^{2} \) |
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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
−8.446705359592458470221337995254, −7.71842564282671087228229068734, −7.25744909578678007692507356062, −6.30894130383397227787338745420, −5.53893140319348772045378124037, −4.47666975985369040199385050444, −3.83806499551894603091974974994, −3.03981863528674724671753240693, −1.58841949748359453222856956012, −0.72185802712059522852093345145,
0.842255369712102166277479483262, 1.96282458015110219835910934878, 3.21624449893756218803946431810, 3.65577832780866913053879336891, 4.94435704896965318370948355780, 5.83968402995863797795193239069, 6.07799097539889760771469862983, 7.31693112157006705252307946560, 8.136591476148417482375200861366, 8.570360090142235469962100327647