| L(s) = 1 | + (0.112 − 1.40i)2-s + (0.0980 − 0.995i)3-s + (−1.97 − 0.316i)4-s + (−3.03 + 1.61i)5-s + (−1.39 − 0.249i)6-s + (0.773 + 3.88i)7-s + (−0.667 + 2.74i)8-s + (−0.980 − 0.195i)9-s + (1.94 + 4.45i)10-s + (3.11 − 3.80i)11-s + (−0.508 + 1.93i)12-s + (−1.46 + 2.73i)13-s + (5.56 − 0.654i)14-s + (1.31 + 3.17i)15-s + (3.80 + 1.24i)16-s + (−1.58 + 3.81i)17-s + ⋯ |
| L(s) = 1 | + (0.0793 − 0.996i)2-s + (0.0565 − 0.574i)3-s + (−0.987 − 0.158i)4-s + (−1.35 + 0.724i)5-s + (−0.568 − 0.101i)6-s + (0.292 + 1.46i)7-s + (−0.235 + 0.971i)8-s + (−0.326 − 0.0650i)9-s + (0.614 + 1.40i)10-s + (0.940 − 1.14i)11-s + (−0.146 + 0.558i)12-s + (−0.406 + 0.759i)13-s + (1.48 − 0.174i)14-s + (0.339 + 0.819i)15-s + (0.950 + 0.312i)16-s + (−0.383 + 0.925i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.657904 + 0.229581i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.657904 + 0.229581i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.112 + 1.40i)T \) |
| 3 | \( 1 + (-0.0980 + 0.995i)T \) |
| good | 5 | \( 1 + (3.03 - 1.61i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.773 - 3.88i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-3.11 + 3.80i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.46 - 2.73i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (1.58 - 3.81i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.563 - 1.85i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (5.28 - 3.53i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-3.53 + 2.89i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (6.59 - 6.59i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.317 + 0.0963i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-0.356 - 0.533i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.577 - 5.86i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (5.75 + 2.38i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (7.52 + 6.17i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-3.25 - 6.08i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-8.53 - 0.841i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-6.73 - 0.663i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-16.2 + 3.23i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (0.0908 - 0.456i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (8.39 - 3.47i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (14.8 - 4.50i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-4.76 - 3.18i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (2.35 - 2.35i)T - 97iT^{2} \) |
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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
−11.56305290698078436742583080279, −11.04753462027955396422244833080, −9.602000469729358435689080784155, −8.519060777704641234800759589201, −8.154997278940025714432022688289, −6.64598396199176174086335010615, −5.60157367285459771991450976121, −4.05191834719933318494097306192, −3.18465850731180580145293531333, −1.85407703461854058462138338319,
0.46447751564508370559168218849, 3.77502833774232628923438133097, 4.35889207570493590816554733272, 5.02910911212663939333786245985, 6.81733799918351474689474418302, 7.51424003660005389069260469014, 8.206208123046083067195171678268, 9.294549575549295447601943119867, 10.09416300034364666869930571726, 11.26588526061042515151402536090
