L(s) = 1 | + (−1.30 + 0.951i)3-s + (−0.587 − 0.809i)5-s + i·7-s + (0.500 − 1.53i)9-s + (−0.309 − 0.951i)11-s + (0.951 + 0.309i)13-s + (1.53 + 0.5i)15-s + (−1.30 − 0.951i)17-s + (−0.5 − 0.363i)19-s + (−0.951 − 1.30i)21-s + (0.951 − 0.309i)23-s + (−0.309 + 0.951i)25-s + (0.309 + 0.951i)27-s + (0.587 + 0.809i)29-s + (0.587 − 0.809i)31-s + ⋯ |
L(s) = 1 | + (−1.30 + 0.951i)3-s + (−0.587 − 0.809i)5-s + i·7-s + (0.500 − 1.53i)9-s + (−0.309 − 0.951i)11-s + (0.951 + 0.309i)13-s + (1.53 + 0.5i)15-s + (−1.30 − 0.951i)17-s + (−0.5 − 0.363i)19-s + (−0.951 − 1.30i)21-s + (0.951 − 0.309i)23-s + (−0.309 + 0.951i)25-s + (0.309 + 0.951i)27-s + (0.587 + 0.809i)29-s + (0.587 − 0.809i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5785399692\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5785399692\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.587 + 0.809i)T \) |
good | 3 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
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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
−9.405011392696965200288382104280, −8.915862771068991011644033077321, −8.349320590885608509007183405913, −6.94107711635354704965862489687, −6.06139093719070633614564303696, −5.43645417122632358300967729752, −4.67770954302149974774492513223, −4.03708820793113223992940035761, −2.70670931210077161827627223170, −0.69715855321432090434568462940,
1.04074018930617883796273314645, 2.37938867806172167380399263916, 3.89629286210148279445147887167, 4.57729930043537049581319044659, 5.81456939005521176085327271372, 6.60431649807921335893372775602, 6.97311726755182227979998665742, 7.74857394998535672768343501178, 8.515126702346343955746390319966, 10.02357230254886395754964909354