L(s) = 1 | + (−1 − i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (−0.224 − 2.22i)5-s + 2.44·6-s + (−3.44 − 3.44i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−2 + 2.44i)10-s − 1.55·11-s + (−2.44 − 2.44i)12-s + 6.89i·14-s + (2.99 + 2.44i)15-s − 4·16-s + (−2.99 + 2.99i)18-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (−0.100 − 0.994i)5-s + 0.999·6-s + (−1.30 − 1.30i)7-s + (0.707 − 0.707i)8-s − 0.999i·9-s + (−0.632 + 0.774i)10-s − 0.467·11-s + (−0.707 − 0.707i)12-s + 1.84i·14-s + (0.774 + 0.632i)15-s − 16-s + (−0.707 + 0.707i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.113642 - 0.334915i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.113642 - 0.334915i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 + (0.224 + 2.22i)T \) |
good | 7 | \( 1 + (3.44 + 3.44i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.55T + 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 5.34iT - 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (2.44 - 2.44i)T - 53iT^{2} \) |
| 59 | \( 1 + 15.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-11.8 + 11.8i)T - 73iT^{2} \) |
| 79 | \( 1 + 14.6iT - 79T^{2} \) |
| 83 | \( 1 + (-4 + 4i)T - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-8.79 - 8.79i)T + 97iT^{2} \) |
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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
−12.84094295954931632712174178585, −11.94380057154034893036654790062, −10.76510175032595696985505584993, −9.952007263840718969119806461024, −9.284810967207230374073333883891, −7.83810298226112589553678994665, −6.43854356893763885829647408399, −4.60347384478523713364864760544, −3.53363303668574277109226204230, −0.50524988570454925015104953906,
2.57727608679077880265242898249, 5.47280225280286794693353628525, 6.33779383600398662852933774617, 7.10133167012079581071248476374, 8.383875159524007905318653143328, 9.702696998792947270740110740771, 10.64359218870153862687237937803, 11.74789055255361908115407160818, 12.82990784968328611677901963518, 13.95398847255071077782654094180