| L(s) = 1 | + (0.726 − 1.21i)2-s + (−0.594 − 1.62i)3-s + (−0.943 − 1.76i)4-s + (−0.0821 − 0.834i)5-s + (−2.40 − 0.461i)6-s + (0.978 + 0.653i)7-s + (−2.82 − 0.135i)8-s + (−2.29 + 1.93i)9-s + (−1.07 − 0.506i)10-s + (−1.45 − 2.72i)11-s + (−2.30 + 2.58i)12-s + (0.457 − 4.64i)13-s + (1.50 − 0.712i)14-s + (−1.30 + 0.629i)15-s + (−2.21 + 3.32i)16-s + (−2.49 + 6.02i)17-s + ⋯ |
| L(s) = 1 | + (0.513 − 0.857i)2-s + (−0.343 − 0.939i)3-s + (−0.471 − 0.881i)4-s + (−0.0367 − 0.373i)5-s + (−0.982 − 0.188i)6-s + (0.369 + 0.247i)7-s + (−0.998 − 0.0480i)8-s + (−0.764 + 0.644i)9-s + (−0.338 − 0.160i)10-s + (−0.439 − 0.822i)11-s + (−0.666 + 0.745i)12-s + (0.127 − 1.28i)13-s + (0.402 − 0.190i)14-s + (−0.337 + 0.162i)15-s + (−0.554 + 0.832i)16-s + (−0.605 + 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0873928 + 1.26148i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0873928 + 1.26148i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.726 + 1.21i)T \) |
| 3 | \( 1 + (0.594 + 1.62i)T \) |
| good | 5 | \( 1 + (0.0821 + 0.834i)T + (-4.90 + 0.975i)T^{2} \) |
| 7 | \( 1 + (-0.978 - 0.653i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (1.45 + 2.72i)T + (-6.11 + 9.14i)T^{2} \) |
| 13 | \( 1 + (-0.457 + 4.64i)T + (-12.7 - 2.53i)T^{2} \) |
| 17 | \( 1 + (2.49 - 6.02i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-3.16 + 2.59i)T + (3.70 - 18.6i)T^{2} \) |
| 23 | \( 1 + (-0.368 + 1.85i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (3.90 - 7.30i)T + (-16.1 - 24.1i)T^{2} \) |
| 31 | \( 1 + (-0.0671 - 0.0671i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.480 + 0.394i)T + (7.21 + 36.2i)T^{2} \) |
| 41 | \( 1 + (-1.26 + 6.33i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-3.03 + 9.99i)T + (-35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-4.56 + 11.0i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.544 + 1.01i)T + (-29.4 + 44.0i)T^{2} \) |
| 59 | \( 1 + (0.557 + 5.66i)T + (-57.8 + 11.5i)T^{2} \) |
| 61 | \( 1 + (-2.81 - 9.28i)T + (-50.7 + 33.8i)T^{2} \) |
| 67 | \( 1 + (-8.24 + 2.50i)T + (55.7 - 37.2i)T^{2} \) |
| 71 | \( 1 + (-4.69 - 3.13i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-0.764 - 1.14i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (5.43 - 2.25i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (5.05 - 4.14i)T + (16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (-5.57 + 1.10i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (6.43 - 6.43i)T - 97iT^{2} \) |
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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
−10.85250101954925560181444685331, −10.55759310366529295135420835356, −8.801154999939645111986127091962, −8.348197097963839641067768980781, −6.91688839553513610548616582595, −5.62329109735929644305691578096, −5.22334123786774051735129555686, −3.51339773458521932607938202507, −2.23106869647259183779915933307, −0.75994278426177479815530189786,
2.88495503410689556712690422955, 4.29281888211440499624902193552, 4.81656850305316171727407923404, 6.00615760742899830311881562064, 7.02427839233221193479038166499, 7.86987547236689071464229205863, 9.278736992799343412073394579557, 9.665205835078442167552267870126, 11.19135954405629498288400177610, 11.60465335653247934298919501302
