L(s) = 1 | + (−0.623 + 1.07i)2-s + (−0.277 − 0.480i)4-s + (−0.5 − 0.866i)7-s − 0.554·8-s + (−0.5 − 0.866i)9-s + (0.222 − 0.385i)11-s + 1.24·14-s + (0.623 − 1.07i)16-s + 1.24·18-s + (0.277 + 0.480i)22-s + (0.900 − 1.56i)23-s + 25-s + (−0.277 + 0.480i)28-s + (−0.623 + 1.07i)29-s + (0.500 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (−0.623 + 1.07i)2-s + (−0.277 − 0.480i)4-s + (−0.5 − 0.866i)7-s − 0.554·8-s + (−0.5 − 0.866i)9-s + (0.222 − 0.385i)11-s + 1.24·14-s + (0.623 − 1.07i)16-s + 1.24·18-s + (0.277 + 0.480i)22-s + (0.900 − 1.56i)23-s + 25-s + (−0.277 + 0.480i)28-s + (−0.623 + 1.07i)29-s + (0.500 + 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6046428884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6046428884\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.582822544027801345901615440034, −8.988437338629833400006900637370, −8.375117864087706295052881642736, −7.33464813278295762641199874663, −6.71558713175034376079722134442, −6.18126054697442499452007278869, −5.08854729075310595449900778446, −3.75403825897156642798294605088, −2.91549534127026354251162891927, −0.67460351321824723772964645058,
1.55758082602634536846886157221, 2.62741868770318602662041303307, 3.30633538852180643121480777003, 4.81967325135694429343913774321, 5.73892012087324760170508769521, 6.59909551827337460004370783247, 7.84625411006289113529584138965, 8.579084329668545966896331796351, 9.515001597879942265988664774985, 9.745303281706286347218317692668