L(s) = 1 | + (0.707 − 0.707i)2-s + (0.292 − 1.70i)3-s − 1.00i·4-s + (−0.707 + 2.12i)5-s + (−0.999 − 1.41i)6-s + (−3 − 3i)7-s + (−0.707 − 0.707i)8-s + (−2.82 − i)9-s + (0.999 + 2i)10-s − 1.41i·11-s + (−1.70 − 0.292i)12-s + (−2 + 2i)13-s − 4.24·14-s + (3.41 + 1.82i)15-s − 1.00·16-s + (−0.707 + 0.707i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.169 − 0.985i)3-s − 0.500i·4-s + (−0.316 + 0.948i)5-s + (−0.408 − 0.577i)6-s + (−1.13 − 1.13i)7-s + (−0.250 − 0.250i)8-s + (−0.942 − 0.333i)9-s + (0.316 + 0.632i)10-s − 0.426i·11-s + (−0.492 − 0.0845i)12-s + (−0.554 + 0.554i)13-s − 1.13·14-s + (0.881 + 0.472i)15-s − 0.250·16-s + (−0.171 + 0.171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0349625 - 1.12918i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0349625 - 1.12918i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.292 + 1.70i)T \) |
| 5 | \( 1 + (0.707 - 2.12i)T \) |
| 17 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + (3 + 3i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + (2 - 2i)T - 13iT^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + (2 + 2i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.48iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (2.82 - 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.65 - 5.65i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + (-10 - 10i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.3iT - 71T^{2} \) |
| 73 | \( 1 + (3 - 3i)T - 73iT^{2} \) |
| 79 | \( 1 + 14iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−10.65097001300400552525001082117, −9.813403364338584597918642420865, −8.696826708120855493386382771360, −7.30421339375458048749440212658, −6.85261370781406499158019558396, −6.13915091699981493125103714929, −4.44738002019127870663619622683, −3.27113013812821530091337727714, −2.52926617754969113857136637879, −0.53424918103776138478706989268,
2.67859133565610921714790538138, 3.73413770322209102508212922207, 4.81953374784323550765130951960, 5.56272523285309459113258261011, 6.46021880493944077043180218039, 8.075408458950135062980248563009, 8.519705286139752625561745608488, 9.700553157072122388932490185858, 9.992592392349487341500001118567, 11.65996609062362033171880452393