L(s) = 1 | + (−1 + i)2-s + (−1 − i)3-s − 2i·4-s + (−2 − i)5-s + 2·6-s + (−2 + 2i)7-s + (2 + 2i)8-s − i·9-s + (3 − i)10-s + (−2 + 2i)12-s − 4i·14-s + (1 + 3i)15-s − 4·16-s + (5 + 5i)17-s + (1 + i)18-s − 19-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.577 − 0.577i)3-s − i·4-s + (−0.894 − 0.447i)5-s + 0.816·6-s + (−0.755 + 0.755i)7-s + (0.707 + 0.707i)8-s − 0.333i·9-s + (0.948 − 0.316i)10-s + (−0.577 + 0.577i)12-s − 1.06i·14-s + (0.258 + 0.774i)15-s − 16-s + (1.21 + 1.21i)17-s + (0.235 + 0.235i)18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0898 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.275250 + 0.301186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.275250 + 0.301186i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 5 | \( 1 + (2 + i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + (2 - 2i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (-5 - 5i)T + 17iT^{2} \) |
| 23 | \( 1 + (-4 - 4i)T + 23iT^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + (6 + 6i)T + 43iT^{2} \) |
| 47 | \( 1 + (2 - 2i)T - 47iT^{2} \) |
| 53 | \( 1 + (10 - 10i)T - 53iT^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (-3 + 3i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + (-4 - 4i)T + 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + (10 + 10i)T + 97iT^{2} \) |
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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
−11.63208236816003624210742197675, −10.66514855508434260980175923208, −9.494366261446820918221325504792, −8.766324518902652080421493177813, −7.81273941244522282525299847116, −6.91556295480652558461808133889, −6.01989445821401619339399391226, −5.19787086294410104352020598983, −3.49799430073330169701001365647, −1.25119970149498671791291730699,
0.42470211655815317100259624463, 2.85659164645844135043673340268, 3.85947417847140959498101092628, 4.88586476572418511361912691176, 6.64130475966195335784825733303, 7.49058586675796400849748712935, 8.340766529103677760043828393316, 9.803573576984259043241855171986, 10.11634113022597540682281333749, 11.15639221790371607001138036526