| L(s) = 1 | + (−2.72 + 0.359i)2-s + (−0.373 − 0.758i)3-s + (5.37 − 1.44i)4-s + (−0.751 + 0.659i)5-s + (1.29 + 1.93i)6-s + (2.64 + 0.169i)7-s + (−9.06 + 3.75i)8-s + (1.39 − 1.81i)9-s + (1.81 − 2.06i)10-s + (3.10 − 0.203i)11-s + (−3.10 − 3.53i)12-s + (1.68 + 1.68i)13-s + (−7.26 + 0.485i)14-s + (0.781 + 0.323i)15-s + (13.7 − 7.93i)16-s + (−1.41 + 3.87i)17-s + ⋯ |
| L(s) = 1 | + (−1.92 + 0.253i)2-s + (−0.215 − 0.437i)3-s + (2.68 − 0.720i)4-s + (−0.336 + 0.294i)5-s + (0.527 + 0.789i)6-s + (0.997 + 0.0640i)7-s + (−3.20 + 1.32i)8-s + (0.463 − 0.604i)9-s + (0.573 − 0.654i)10-s + (0.935 − 0.0613i)11-s + (−0.895 − 1.02i)12-s + (0.467 + 0.467i)13-s + (−1.94 + 0.129i)14-s + (0.201 + 0.0835i)15-s + (3.43 − 1.98i)16-s + (−0.343 + 0.938i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.673593 + 0.0202349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.673593 + 0.0202349i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.64 - 0.169i)T \) |
| 17 | \( 1 + (1.41 - 3.87i)T \) |
| good | 2 | \( 1 + (2.72 - 0.359i)T + (1.93 - 0.517i)T^{2} \) |
| 3 | \( 1 + (0.373 + 0.758i)T + (-1.82 + 2.38i)T^{2} \) |
| 11 | \( 1 + (-3.10 + 0.203i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-1.68 - 1.68i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.128 + 0.973i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-2.91 - 1.43i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-0.293 - 1.47i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (6.02 - 2.97i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.508 + 7.75i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (0.615 - 3.09i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-3.01 - 7.27i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-2.25 + 8.42i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (3.80 + 4.95i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (-10.8 - 1.42i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-2.66 - 7.83i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-7.94 - 4.58i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.0 - 8.06i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-4.93 - 1.67i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-1.28 + 2.60i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-4.48 + 10.8i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (4.60 + 1.23i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.19 + 0.437i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.70929749125248331785404956883, −9.581557529729495034024932426084, −8.852584486055179971352782395572, −8.199297380506077968652460587572, −7.13484311900993846295639145961, −6.76189073222227709588696799920, −5.72111755902463828123197508924, −3.81410009507584242334187656172, −1.98228413099777407520269123321, −1.07604763130577113359800027935,
0.986527043237504972847705642480, 2.17975618289971981480929156653, 3.81274849013213193954490327265, 5.18172984046996145922702354710, 6.60757334190026946866349584155, 7.53455827041790569996134021117, 8.148646423276548749732979805833, 9.028088958512772419850230894329, 9.662375884824823503413248063974, 10.71239148254602696583846833742
