L(s) = 1 | + (−2.99 − 5.19i)5-s + (−0.375 − 18.5i)7-s + (39.3 + 22.7i)11-s + (−22.7 + 13.1i)13-s + 19.7·17-s − 27.9i·19-s + (60.3 − 34.8i)23-s + (44.5 − 77.0i)25-s + (119. + 68.7i)29-s + (138. − 79.7i)31-s + (−95.0 + 57.4i)35-s − 287.·37-s + (20.4 + 35.3i)41-s + (55.4 − 95.9i)43-s + (−109. + 189. i)47-s + ⋯ |
L(s) = 1 | + (−0.268 − 0.464i)5-s + (−0.0202 − 0.999i)7-s + (1.07 + 0.623i)11-s + (−0.485 + 0.280i)13-s + 0.281·17-s − 0.337i·19-s + (0.547 − 0.315i)23-s + (0.356 − 0.616i)25-s + (0.762 + 0.440i)29-s + (0.800 − 0.462i)31-s + (−0.459 + 0.277i)35-s − 1.27·37-s + (0.0778 + 0.134i)41-s + (0.196 − 0.340i)43-s + (−0.340 + 0.589i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 + 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.247 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.634896051\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.634896051\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.375 + 18.5i)T \) |
good | 5 | \( 1 + (2.99 + 5.19i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-39.3 - 22.7i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (22.7 - 13.1i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 19.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-60.3 + 34.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-119. - 68.7i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-138. + 79.7i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 287.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-20.4 - 35.3i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-55.4 + 95.9i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (109. - 189. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 209. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (413. + 716. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (594. + 343. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (171. + 296. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 387. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 220. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-242. + 419. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-354. + 613. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 140.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.30e3 + 753. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−9.706909680962453338824707755372, −8.898012740269084243840230431685, −7.956185313326976557675874790938, −7.01968476410284290862581722843, −6.43162040194403195507064452864, −4.84706342560675164313739187157, −4.36914937479045074512386291335, −3.20847822584943787671087025291, −1.62343863522668128806883589209, −0.48973607896895457011897803504,
1.25553673798383395478376855983, 2.72679709293993760298002977419, 3.53700569483824358794874194798, 4.85032478824373498366141554108, 5.84754431856934146331149176492, 6.64086831078480153266951411919, 7.60494893789493060442663420886, 8.625901712113704589869032611934, 9.206999798983143649296089557361, 10.21705075257007703692556001533