L(s) = 1 | + (−0.366 + 1.36i)2-s + 1.73·3-s + (−1.73 − i)4-s + (−0.633 + 2.36i)6-s + (1.73 + 2i)7-s + (2 − 1.99i)8-s − 0.267i·11-s + (−2.99 − 1.73i)12-s + 0.464i·13-s + (−3.36 + 1.63i)14-s + (1.99 + 3.46i)16-s + 6.46i·17-s + 6·19-s + (2.99 + 3.46i)21-s + (0.366 + 0.0980i)22-s + 1.46i·23-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + 1.00·3-s + (−0.866 − 0.5i)4-s + (−0.258 + 0.965i)6-s + (0.654 + 0.755i)7-s + (0.707 − 0.707i)8-s − 0.0807i·11-s + (−0.866 − 0.499i)12-s + 0.128i·13-s + (−0.899 + 0.436i)14-s + (0.499 + 0.866i)16-s + 1.56i·17-s + 1.37·19-s + (0.654 + 0.755i)21-s + (0.0780 + 0.0209i)22-s + 0.305i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11575 + 1.35095i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11575 + 1.35095i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 11 | \( 1 + 0.267iT - 11T^{2} \) |
| 13 | \( 1 - 0.464iT - 13T^{2} \) |
| 17 | \( 1 - 6.46iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 1.46iT - 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 9.46T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 1.73T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 9.46iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 7.46iT - 71T^{2} \) |
| 73 | \( 1 + 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 14.6iT - 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2.53iT - 89T^{2} \) |
| 97 | \( 1 - 13.3iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.41078933293567975975092713160, −9.381997318406651968004706958065, −8.840180632416924744407757954255, −8.067789849262556164493590586349, −7.57044891783568918210988783292, −6.21501182523023839745274184099, −5.48761035990834622754531498287, −4.35005406136056641150195919797, −3.13490689453419752830703271468, −1.64720153014001527606567993930,
1.01518454152294140742474394079, 2.49704480360999234001562125660, 3.28058496071971325801247671847, 4.39834078297719984233271961274, 5.30854930963982759558531380329, 7.16543114526691949933733476125, 7.84903429908166203145550050108, 8.573748920913104722376014385567, 9.512250599056241966927980142268, 9.991004052922898728737488012200