| L(s) = 1 | + (−0.951 + 0.309i)2-s + (−1.05 + 1.36i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (0.584 − 1.63i)6-s + (0.140 + 1.34i)7-s + (−0.587 + 0.809i)8-s + (−0.753 − 2.90i)9-s + (0.669 − 0.743i)10-s + (3.37 + 1.50i)11-s + (−0.0522 + 1.73i)12-s + (−1.36 − 6.41i)13-s + (−0.548 − 1.23i)14-s + (0.232 − 1.71i)15-s + (0.309 − 0.951i)16-s + (−5.12 + 2.28i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.611 + 0.790i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (0.238 − 0.665i)6-s + (0.0532 + 0.506i)7-s + (−0.207 + 0.286i)8-s + (−0.251 − 0.967i)9-s + (0.211 − 0.235i)10-s + (1.01 + 0.452i)11-s + (−0.0150 + 0.499i)12-s + (−0.377 − 1.77i)13-s + (−0.146 − 0.329i)14-s + (0.0601 − 0.443i)15-s + (0.0772 − 0.237i)16-s + (−1.24 + 0.553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.473279 - 0.182206i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.473279 - 0.182206i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (1.05 - 1.36i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (-4.68 + 3.01i)T \) |
| good | 7 | \( 1 + (-0.140 - 1.34i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-3.37 - 1.50i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (1.36 + 6.41i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (5.12 - 2.28i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (1.80 + 0.383i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (4.04 + 2.93i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.419 + 1.29i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (3.11 + 1.79i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.81 + 1.63i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.02 + 9.52i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-2.26 - 0.737i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.959 - 9.13i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-7.31 + 6.58i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 1.70iT - 61T^{2} \) |
| 67 | \( 1 + (0.0499 + 0.0865i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.04 + 0.845i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-1.50 + 3.37i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-0.331 - 0.745i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-3.07 + 3.41i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-9.31 + 6.77i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-13.8 + 10.0i)T + (29.9 - 92.2i)T^{2} \) |
| 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
−10.22665173042721335490457875794, −9.055925690721929254323284486638, −8.541928001819442321966244086996, −7.47104294477569923975833319348, −6.43877843709211228092237309866, −5.81504405374442253150896041895, −4.68851026041871635578186095736, −3.73225656102741705976573811603, −2.34796214108631663917630752649, −0.36044711764152058366346690587,
1.18125491374051476461532887054, 2.23882249405392604253045248146, 3.92586395719615269530222376219, 4.78258591720507131270785774227, 6.33693454168630775874428479004, 6.77215635716596719601474328273, 7.55412532392411855071461480565, 8.580366408545140153121773292018, 9.180372722169548375355891127148, 10.22441473297331612812681584861
