L(s) = 1 | + (2.31 + 0.619i)2-s + (0.866 + 1.5i)3-s + (3.23 + 1.86i)4-s + (−1.23 + 1.23i)5-s + (1.07 + 4.00i)6-s + (2.93 + 2.93i)8-s + (−1.5 + 2.59i)9-s + (−3.63 + 2.09i)10-s + (1.69 − 6.31i)11-s + 6.46i·12-s + (−2.93 − 0.785i)15-s + (1.23 + 2.13i)16-s + (−5.07 + 5.07i)18-s + (−6.31 + 1.69i)20-s + (7.83 − 13.5i)22-s + ⋯ |
L(s) = 1 | + (1.63 + 0.438i)2-s + (0.499 + 0.866i)3-s + (1.61 + 0.933i)4-s + (−0.554 + 0.554i)5-s + (0.438 + 1.63i)6-s + (1.03 + 1.03i)8-s + (−0.5 + 0.866i)9-s + (−1.14 + 0.663i)10-s + (0.510 − 1.90i)11-s + 1.86i·12-s + (−0.757 − 0.202i)15-s + (0.308 + 0.533i)16-s + (−1.19 + 1.19i)18-s + (−1.41 + 0.378i)20-s + (1.66 − 2.89i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.63480 + 2.56771i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.63480 + 2.56771i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 1.5i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-2.31 - 0.619i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (1.23 - 1.23i)T - 5iT^{2} \) |
| 7 | \( 1 + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.69 + 6.31i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31iT^{2} \) |
| 37 | \( 1 + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-7.55 - 2.02i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.46 - 2i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (7.10 + 7.10i)T + 47iT^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-0.453 + 0.121i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (6.92 - 12i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (4.17 + 15.5i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + (8.91 - 8.91i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.11 - 4.17i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (84.0 - 48.5i)T^{2} \) |
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
−11.28430140420076721228729271090, −10.65672153698792997598191768812, −9.242850461847932755091505336326, −8.285946222846588377076207628120, −7.31729298990015027633031830552, −6.17552889206538220450024168751, −5.43986292700352144443774142421, −4.24515812705287183770135197555, −3.51062001394689198702561074938, −2.83894836680697023088072432065,
1.61952475841461065483550852555, 2.69961439291851440483029780046, 4.03031172764171224342283785422, 4.64173271410630177290949721004, 5.92283952081654133871477847041, 6.91066093628545263928937303087, 7.65196693298823977001002321826, 8.865994799991644726523298652083, 9.913107181642176639511117609924, 11.24995224188582753994626894837