L(s) = 1 | + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.118 + 0.822i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.544 − 1.19i)11-s + (−0.239 − 1.66i)13-s + (0.698 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (0.841 − 0.540i)18-s + (1.25 + 1.45i)19-s + (−0.959 − 0.281i)20-s − 1.30·22-s + (−0.959 − 0.281i)23-s + ⋯ |
L(s) = 1 | + (0.415 − 0.909i)2-s + (−0.654 − 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.118 + 0.822i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.142 − 0.989i)10-s + (−0.544 − 1.19i)11-s + (−0.239 − 1.66i)13-s + (0.698 + 0.449i)14-s + (−0.142 + 0.989i)16-s + (0.841 − 0.540i)18-s + (1.25 + 1.45i)19-s + (−0.959 − 0.281i)20-s − 1.30·22-s + (−0.959 − 0.281i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.291058075\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.291058075\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (0.959 + 0.281i)T \) |
good | 3 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \) |
| 11 | \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 17 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 37 | \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 43 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 + 0.284T + T^{2} \) |
| 53 | \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
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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
−10.20319185007286293180112835216, −9.549555797959036775065284840322, −8.472112882197336248335172536126, −7.87440758416569947212738162012, −6.03134693695241862177977687681, −5.59409455726080558168881956070, −4.90833150765758889822241213022, −3.46171556743462380009254772990, −2.55588980427917629181650139727, −1.32563768468995406613137229090,
1.99096557536213872472456854037, 3.48805675900816268410049136050, 4.47671088188292422564696184181, 5.21497456682079929288776634136, 6.58613009174313752327583325132, 6.98767708829989954818279294400, 7.46458899655945098744702046754, 8.945328428640206388884698155169, 9.752521860039209790105001849014, 10.04508802978105864823515751546