| L(s) = 1 | + 4i·3-s + 8·4-s + 22i·7-s + 11·9-s − 12·11-s + 32i·12-s + 8i·13-s + 64·16-s + 66i·17-s − 19·19-s − 88·21-s − 30i·23-s + 152i·27-s + 176i·28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.769i·3-s + 4-s + 1.18i·7-s + 0.407·9-s − 0.328·11-s + 0.769i·12-s + 0.170i·13-s + 16-s + 0.941i·17-s − 0.229·19-s − 0.914·21-s − 0.271i·23-s + 1.08i·27-s + 1.18i·28-s + 0.0384·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.382968969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.382968969\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 8T^{2} \) |
| 3 | \( 1 - 4iT - 27T^{2} \) |
| 7 | \( 1 - 22iT - 343T^{2} \) |
| 11 | \( 1 + 12T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 66iT - 4.91e3T^{2} \) |
| 23 | \( 1 + 30iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 64T + 2.97e4T^{2} \) |
| 37 | \( 1 - 16iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 54T + 6.89e4T^{2} \) |
| 43 | \( 1 - 182iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 594iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 396iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 564T + 2.05e5T^{2} \) |
| 61 | \( 1 + 706T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 984T + 3.57e5T^{2} \) |
| 73 | \( 1 - 14iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 328T + 4.93e5T^{2} \) |
| 83 | \( 1 + 294iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 918T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.56e3iT - 9.12e5T^{2} \) |
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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
−10.73966274422633497508161076919, −10.18406503807177439193290184694, −9.146715459004593933567201295568, −8.271146336966628212226595544025, −7.16057768150811965415158955758, −6.13051214325580483293007110410, −5.31104677877566389281339034035, −4.03803826693056756335497709082, −2.81656264705695824356608354557, −1.72771380708012989216226556933,
0.72561941375950101491173175709, 1.84514794228706342810821136480, 3.11731659695643039616468365006, 4.43933712848110202473678035981, 5.84503165970060075833455934852, 6.93265438905431378923307733992, 7.32327552474841525912567636267, 8.086426824926475821269573942101, 9.612386621062663337360349453925, 10.47186974879201282410007471784
