L(s) = 1 | + (−0.309 + 0.951i)4-s + (0.503 + 1.27i)7-s + (1.15 − 1.23i)13-s + (−0.809 − 0.587i)16-s + (−0.393 + 1.21i)19-s + (0.425 + 0.904i)25-s + (−1.36 + 0.0859i)28-s + (−1.80 + 0.462i)31-s + (1.27 + 1.19i)37-s + (−1.84 − 0.730i)43-s + (−0.637 + 0.598i)49-s + (0.813 + 1.47i)52-s + (0.566 − 1.03i)61-s + (0.809 − 0.587i)64-s + (1.65 − 1.05i)67-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)4-s + (0.503 + 1.27i)7-s + (1.15 − 1.23i)13-s + (−0.809 − 0.587i)16-s + (−0.393 + 1.21i)19-s + (0.425 + 0.904i)25-s + (−1.36 + 0.0859i)28-s + (−1.80 + 0.462i)31-s + (1.27 + 1.19i)37-s + (−1.84 − 0.730i)43-s + (−0.637 + 0.598i)49-s + (0.813 + 1.47i)52-s + (0.566 − 1.03i)61-s + (0.809 − 0.587i)64-s + (1.65 − 1.05i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.048636448\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.048636448\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 151 | \( 1 + (-0.809 - 0.587i)T \) |
good | 2 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.425 - 0.904i)T^{2} \) |
| 7 | \( 1 + (-0.503 - 1.27i)T + (-0.728 + 0.684i)T^{2} \) |
| 11 | \( 1 + (0.535 + 0.844i)T^{2} \) |
| 13 | \( 1 + (-1.15 + 1.23i)T + (-0.0627 - 0.998i)T^{2} \) |
| 17 | \( 1 + (0.728 + 0.684i)T^{2} \) |
| 19 | \( 1 + (0.393 - 1.21i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (0.968 - 0.248i)T^{2} \) |
| 31 | \( 1 + (1.80 - 0.462i)T + (0.876 - 0.481i)T^{2} \) |
| 37 | \( 1 + (-1.27 - 1.19i)T + (0.0627 + 0.998i)T^{2} \) |
| 41 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 43 | \( 1 + (1.84 + 0.730i)T + (0.728 + 0.684i)T^{2} \) |
| 47 | \( 1 + (-0.637 - 0.770i)T^{2} \) |
| 53 | \( 1 + (0.929 + 0.368i)T^{2} \) |
| 59 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.566 + 1.03i)T + (-0.535 - 0.844i)T^{2} \) |
| 67 | \( 1 + (-1.65 + 1.05i)T + (0.425 - 0.904i)T^{2} \) |
| 71 | \( 1 + (-0.728 + 0.684i)T^{2} \) |
| 73 | \( 1 + (-0.700 - 1.76i)T + (-0.728 + 0.684i)T^{2} \) |
| 79 | \( 1 + (-0.450 + 1.75i)T + (-0.876 - 0.481i)T^{2} \) |
| 83 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 89 | \( 1 + (0.187 + 0.982i)T^{2} \) |
| 97 | \( 1 + (0.866 + 1.36i)T + (-0.425 + 0.904i)T^{2} \) |
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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
−9.837619589069146926823730780506, −8.926845728994455211143148223458, −8.300171454880429392760733949738, −7.930150472817694616572356329657, −6.72111670020861565872959903320, −5.67053285255051223602853598049, −5.08426539608555124161460442068, −3.74342898562578890673110837796, −3.10273562841066834983885551513, −1.80698452316654623059925013661,
0.968393060239345825321019423699, 2.09882974744794936450499561840, 3.90130324945685428882534560356, 4.40368651258148753323400403724, 5.36179106131313752300098334825, 6.48211382321820732150167474385, 6.93787622829502075168977354000, 8.068688651984587130312327121345, 8.962721161568281854442553658383, 9.555161019566369132297675611933