L(s) = 1 | + (−0.292 + 1.70i)3-s + (−2 − 2i)7-s + (−2.82 − i)9-s − 5.65i·11-s + (−2.82 + 2.82i)17-s − 4i·19-s + (4 − 2.82i)21-s + (−4.24 − 4.24i)23-s + (2.53 − 4.53i)27-s − 5.65·29-s + 8·31-s + (9.65 + 1.65i)33-s + (8 + 8i)37-s − 5.65i·41-s + (−2 + 2i)43-s + ⋯ |
L(s) = 1 | + (−0.169 + 0.985i)3-s + (−0.755 − 0.755i)7-s + (−0.942 − 0.333i)9-s − 1.70i·11-s + (−0.685 + 0.685i)17-s − 0.917i·19-s + (0.872 − 0.617i)21-s + (−0.884 − 0.884i)23-s + (0.487 − 0.872i)27-s − 1.05·29-s + 1.43·31-s + (1.68 + 0.288i)33-s + (1.31 + 1.31i)37-s − 0.883i·41-s + (−0.304 + 0.304i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.514907 - 0.483979i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.514907 - 0.483979i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.292 - 1.70i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (2 + 2i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (2.82 - 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (4.24 + 4.24i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-8 - 8i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (2 - 2i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.41 + 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.65 + 5.65i)T + 53iT^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (6 + 6i)T + 67iT^{2} \) |
| 71 | \( 1 - 11.3iT - 71T^{2} \) |
| 73 | \( 1 + (-8 + 8i)T - 73iT^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + (9.89 + 9.89i)T + 83iT^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + (8 + 8i)T + 97iT^{2} \) |
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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
−10.52112988028441136435944972993, −9.688495470443984327166757642219, −8.791832748349670179108095277248, −8.057712679896400733452429619945, −6.50791660362568492561762371584, −6.03378294820230431158942823606, −4.69356754640700969671639254612, −3.78386079499813411511917271885, −2.87984528571417189245894499157, −0.38447119718784428726770885984,
1.80731057618865983594773881735, 2.78047101899898264838600938747, 4.37009310926899424655781481218, 5.64189303461198336316953931195, 6.38350796570949937163100000521, 7.33084295128124573132817398062, 7.998944319359414530646437006718, 9.301672229605101429192155534231, 9.753997643076284633844079957989, 11.04229972753002009675876511103