| L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.0727 − 0.223i)3-s + (0.309 + 0.951i)4-s + (1.97 − 1.05i)5-s + (0.0727 − 0.223i)6-s + 7-s + (−0.309 + 0.951i)8-s + (2.38 − 1.73i)9-s + (2.21 + 0.308i)10-s + (−2.81 − 2.04i)11-s + (0.190 − 0.138i)12-s + (−0.624 + 0.453i)13-s + (0.809 + 0.587i)14-s + (−0.378 − 0.365i)15-s + (−0.809 + 0.587i)16-s + (0.149 − 0.461i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.0419 − 0.129i)3-s + (0.154 + 0.475i)4-s + (0.882 − 0.470i)5-s + (0.0296 − 0.0913i)6-s + 0.377·7-s + (−0.109 + 0.336i)8-s + (0.794 − 0.576i)9-s + (0.700 + 0.0976i)10-s + (−0.850 − 0.617i)11-s + (0.0549 − 0.0399i)12-s + (−0.173 + 0.125i)13-s + (0.216 + 0.157i)14-s + (−0.0978 − 0.0942i)15-s + (−0.202 + 0.146i)16-s + (0.0363 − 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.07114 + 0.234668i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07114 + 0.234668i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-1.97 + 1.05i)T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + (0.0727 + 0.223i)T + (-2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (2.81 + 2.04i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.624 - 0.453i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.149 + 0.461i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.01 - 6.19i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-6.90 - 5.01i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.84 + 5.67i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.20 - 3.70i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.06 - 0.771i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (5.86 - 4.26i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 7.85T + 43T^{2} \) |
| 47 | \( 1 + (3.48 + 10.7i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.66 - 8.19i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (8.49 - 6.17i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (10.5 + 7.68i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.45 + 7.55i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.0996 + 0.306i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.07 - 4.41i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.331 - 1.01i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.48 - 4.58i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (9.89 + 7.19i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.340 + 1.04i)T + (-78.4 + 57.0i)T^{2} \) |
| 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
−11.76061377254652771567690202350, −10.55666150734487870773291720373, −9.679380429316541689948644211811, −8.615992524114451298664582237371, −7.65530032459656092763198100937, −6.54549013747095900793577831668, −5.60458057141602711228075215176, −4.76595201407456083366681378153, −3.36135940048854413754041389601, −1.67215163130949558656185277572,
1.87237632033751584356684004171, 2.92039649976911951959147763655, 4.69342025450636131201183147132, 5.18336615289120984671745298684, 6.63766219914518756767028562470, 7.39197775714332095193671951182, 8.883621505369860069916044167598, 9.932191579240851452167580248816, 10.65571998262931892190266306921, 11.17664239380288597393049164230
