| L(s) = 1 | + (−1.67 + 0.220i)2-s + (−0.174 − 0.354i)3-s + (0.816 − 0.218i)4-s + (−0.751 + 0.659i)5-s + (0.369 + 0.553i)6-s + (−1.09 + 2.40i)7-s + (1.79 − 0.745i)8-s + (1.73 − 2.25i)9-s + (1.11 − 1.26i)10-s + (−3.64 + 0.239i)11-s + (−0.219 − 0.250i)12-s + (−2.06 − 2.06i)13-s + (1.29 − 4.26i)14-s + (0.364 + 0.151i)15-s + (−4.30 + 2.48i)16-s + (0.843 + 4.03i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 + 0.155i)2-s + (−0.100 − 0.204i)3-s + (0.408 − 0.109i)4-s + (−0.336 + 0.294i)5-s + (0.151 + 0.226i)6-s + (−0.413 + 0.910i)7-s + (0.636 − 0.263i)8-s + (0.577 − 0.752i)9-s + (0.351 − 0.401i)10-s + (−1.10 + 0.0720i)11-s + (−0.0635 − 0.0724i)12-s + (−0.573 − 0.573i)13-s + (0.347 − 1.14i)14-s + (0.0941 + 0.0390i)15-s + (−1.07 + 0.622i)16-s + (0.204 + 0.978i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.319234 - 0.246687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.319234 - 0.246687i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.751 - 0.659i)T \) |
| 7 | \( 1 + (1.09 - 2.40i)T \) |
| 17 | \( 1 + (-0.843 - 4.03i)T \) |
| good | 2 | \( 1 + (1.67 - 0.220i)T + (1.93 - 0.517i)T^{2} \) |
| 3 | \( 1 + (0.174 + 0.354i)T + (-1.82 + 2.38i)T^{2} \) |
| 11 | \( 1 + (3.64 - 0.239i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (2.06 + 2.06i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.347 + 2.63i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-6.16 - 3.04i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (0.847 + 4.26i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-1.66 + 0.821i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.540 + 8.24i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (-2.12 + 10.7i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (0.644 + 1.55i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-1.45 + 5.44i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.60 - 4.69i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (11.2 + 1.47i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (4.40 + 12.9i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (6.33 + 3.65i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.1 - 8.13i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-5.36 - 1.82i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-7.04 + 14.2i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-3.41 + 8.24i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-4.46 - 1.19i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-4.06 + 0.808i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.35357310385837319082472922002, −9.470552625313709046314873101360, −8.870590180728760575939796777936, −7.80317670629840356551446665764, −7.27615410099734479829960350311, −6.20497476518354143436128043812, −5.06327785130807033695845306321, −3.61494984081060860677725140953, −2.24144158718431701876572890612, −0.39280431984616460782305611016,
1.17749725325507117530385321922, 2.82331892924405170301621719095, 4.52319258225564903394490424984, 5.00012441661657320425123779238, 6.83673039282360902212511849868, 7.57677314135977668720120085957, 8.164493030358727000934841482008, 9.326720005933164281893310634971, 9.946038906063066112989757595515, 10.64314587678965263521149468643
