L(s) = 1 | + (−0.909 − 0.415i)2-s + (−0.989 + 0.142i)3-s + (0.654 + 0.755i)4-s + (−0.841 + 0.540i)5-s + (0.959 + 0.281i)6-s + (0.281 + 0.0405i)7-s + (−0.281 − 0.959i)8-s + (0.959 − 0.281i)9-s + (0.989 − 0.142i)10-s + (−0.755 − 0.654i)12-s + (−0.239 − 0.153i)14-s + (0.755 − 0.654i)15-s + (−0.142 + 0.989i)16-s + (−0.989 − 0.142i)18-s + (−0.959 − 0.281i)20-s − 0.284·21-s + ⋯ |
L(s) = 1 | + (−0.909 − 0.415i)2-s + (−0.989 + 0.142i)3-s + (0.654 + 0.755i)4-s + (−0.841 + 0.540i)5-s + (0.959 + 0.281i)6-s + (0.281 + 0.0405i)7-s + (−0.281 − 0.959i)8-s + (0.959 − 0.281i)9-s + (0.989 − 0.142i)10-s + (−0.755 − 0.654i)12-s + (−0.239 − 0.153i)14-s + (0.755 − 0.654i)15-s + (−0.142 + 0.989i)16-s + (−0.989 − 0.142i)18-s + (−0.959 − 0.281i)20-s − 0.284·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4023895617\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4023895617\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.909 + 0.415i)T \) |
| 3 | \( 1 + (0.989 - 0.142i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (-0.909 - 0.415i)T \) |
good | 7 | \( 1 + (-0.281 - 0.0405i)T + (0.959 + 0.281i)T^{2} \) |
| 11 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 13 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 17 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (0.817 + 0.708i)T + (0.142 + 0.989i)T^{2} \) |
| 31 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 37 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 41 | \( 1 + (-1.07 - 1.66i)T + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (0.368 - 1.25i)T + (-0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 - 1.68iT - T^{2} \) |
| 53 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.425 - 1.45i)T + (-0.841 + 0.540i)T^{2} \) |
| 67 | \( 1 + (-1.74 - 0.797i)T + (0.654 + 0.755i)T^{2} \) |
| 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 | \( 1 + (-0.474 - 0.304i)T + (0.415 + 0.909i)T^{2} \) |
| 89 | \( 1 + (-0.797 - 0.234i)T + (0.841 + 0.540i)T^{2} \) |
| 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
−9.921915606850810880195836003410, −9.352643869778715756403689259549, −8.159447143668676310657794578424, −7.57799771340870720086224183989, −6.79333095513847802854053386629, −6.02142681716062448946324122359, −4.71183989374778152671096163088, −3.83184917770763722638149530247, −2.74380866895135514175301858274, −1.17858614188597717100599911505,
0.60974520551767991433453738890, 1.91700299744452216758900488934, 3.73662345259280010911886907635, 4.99083751512361851752597120480, 5.43014306612305318072954635394, 6.64552302292132402537441360164, 7.21815816426445118610180822148, 7.976364340374864580499959010527, 8.797535229652262265663771414929, 9.535137126927674798610893577498