L(s) = 1 | + (−0.550 + 1.30i)2-s + (0.707 − 0.707i)3-s + (−1.39 − 1.43i)4-s + (0.162 + 2.23i)5-s + (0.531 + 1.31i)6-s + 2.93·7-s + (2.63 − 1.02i)8-s − 1.00i·9-s + (−2.99 − 1.01i)10-s + (−0.663 + 0.663i)11-s + (−1.99 − 0.0281i)12-s + (1.12 − 1.12i)13-s + (−1.61 + 3.82i)14-s + (1.69 + 1.46i)15-s + (−0.112 + 3.99i)16-s + 7.47i·17-s + ⋯ |
L(s) = 1 | + (−0.389 + 0.921i)2-s + (0.408 − 0.408i)3-s + (−0.697 − 0.716i)4-s + (0.0724 + 0.997i)5-s + (0.217 + 0.534i)6-s + 1.10·7-s + (0.931 − 0.363i)8-s − 0.333i·9-s + (−0.946 − 0.321i)10-s + (−0.200 + 0.200i)11-s + (−0.577 − 0.00813i)12-s + (0.312 − 0.312i)13-s + (−0.431 + 1.02i)14-s + (0.436 + 0.377i)15-s + (−0.0281 + 0.999i)16-s + 1.81i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.994745 + 0.697059i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.994745 + 0.697059i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.550 - 1.30i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.162 - 2.23i)T \) |
good | 7 | \( 1 - 2.93T + 7T^{2} \) |
| 11 | \( 1 + (0.663 - 0.663i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.12 + 1.12i)T - 13iT^{2} \) |
| 17 | \( 1 - 7.47iT - 17T^{2} \) |
| 19 | \( 1 + (-0.423 - 0.423i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.17T + 23T^{2} \) |
| 29 | \( 1 + (2.95 + 2.95i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 + (5.53 + 5.53i)T + 37iT^{2} \) |
| 41 | \( 1 + 12.3iT - 41T^{2} \) |
| 43 | \( 1 + (-0.897 - 0.897i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.12iT - 47T^{2} \) |
| 53 | \( 1 + (-0.146 - 0.146i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.72 - 7.72i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.37 + 7.37i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.68 + 8.68i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.95iT - 71T^{2} \) |
| 73 | \( 1 - 0.174T + 73T^{2} \) |
| 79 | \( 1 - 3.06T + 79T^{2} \) |
| 83 | \( 1 + (9.18 - 9.18i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.71iT - 89T^{2} \) |
| 97 | \( 1 - 10.5iT - 97T^{2} \) |
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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
−12.49646906257478540580962936448, −10.95655142806496549093768124489, −10.48812829087621706298669307844, −9.115911849813905428916730918088, −8.142910516348132702571438273735, −7.48421637515606916937713425639, −6.45879113390439051943358034935, −5.38967191223490458252991788679, −3.82880314393169191908544099378, −1.83711586504177115011026714741,
1.35612082180812258367801196281, 2.97908467675940621986125093487, 4.57988020374511536861130275970, 5.09573728586129739533575968825, 7.42576009051811669450074125883, 8.416530749993736319254967443198, 9.063139618630296542578464590687, 9.891150913337552788646462821464, 11.20473117920743013008266373920, 11.61399875355017512115558722798