| L(s) = 1 | − 2i·2-s + (4.47 − 2.64i)3-s − 4·4-s − 0.593·5-s + (−5.29 − 8.94i)6-s + 8i·8-s + (12.9 − 23.6i)9-s + 1.18i·10-s + 1.74i·11-s + (−17.8 + 10.5i)12-s − 66.1i·13-s + (−2.65 + 1.57i)15-s + 16·16-s − 37.7·17-s + (−47.3 − 25.9i)18-s − 7.75i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.860 − 0.509i)3-s − 0.5·4-s − 0.0531·5-s + (−0.360 − 0.608i)6-s + 0.353i·8-s + (0.480 − 0.877i)9-s + 0.0375i·10-s + 0.0478i·11-s + (−0.430 + 0.254i)12-s − 1.41i·13-s + (−0.0457 + 0.0270i)15-s + 0.250·16-s − 0.538·17-s + (−0.620 − 0.339i)18-s − 0.0936i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.875161260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875161260\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 + (-4.47 + 2.64i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.593T + 125T^{2} \) |
| 11 | \( 1 - 1.74iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 66.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 37.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 7.75iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 167. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 197. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 300. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 378.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 283.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 314.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 353.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 230. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 573.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 533. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 154.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 135. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 58.5iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 77.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 532.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 6.38iT - 9.12e5T^{2} \) |
| 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
−10.80097371286748313866810873537, −10.07568645247580539148792569777, −8.946198047913932334225862994258, −8.239766381441102802234720767911, −7.24682165528882248355772644684, −5.92490271653501710785736310323, −4.41343804198684002608538392680, −3.20489807667702542126173738619, −2.19100381677799400666900136181, −0.62357053913654366938236399217,
1.94898957210076503675280825771, 3.64054626900765167732645910005, 4.50844374875717433249673999058, 5.78434808702401123374520296375, 7.11373624431249057583834884410, 7.84918152341060488833088361898, 9.116056292747319142250674269733, 9.360808623765655914182848472047, 10.65778744351078994781638137427, 11.70094045350971883671211741574
