L(s) = 1 | + (−0.5 − 0.866i)2-s − 1.41·3-s + (0.500 − 0.866i)4-s + (−1.34 + 2.32i)5-s + (0.707 + 1.22i)6-s − 3·8-s − 0.999·9-s + 2.68·10-s + 5.79·11-s + (−0.707 + 1.22i)12-s + (−2.75 + 2.32i)13-s + (1.89 − 3.28i)15-s + (0.500 + 0.866i)16-s + (2.75 − 4.77i)17-s + (0.499 + 0.866i)18-s + 2.82·19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s − 0.816·3-s + (0.250 − 0.433i)4-s + (−0.600 + 1.03i)5-s + (0.288 + 0.499i)6-s − 1.06·8-s − 0.333·9-s + 0.848·10-s + 1.74·11-s + (−0.204 + 0.353i)12-s + (−0.764 + 0.644i)13-s + (0.490 − 0.848i)15-s + (0.125 + 0.216i)16-s + (0.668 − 1.15i)17-s + (0.117 + 0.204i)18-s + 0.648·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.676422 - 0.468256i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.676422 - 0.468256i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.75 - 2.32i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + (1.34 - 2.32i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 17 | \( 1 + (-2.75 + 4.77i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + (0.897 + 1.55i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.39 + 7.61i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.707 + 1.22i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.39 - 5.88i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.87 + 8.44i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.897 - 1.55i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.29 + 5.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.562 + 0.974i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 - 5.79T + 67T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.90 + 5.02i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.89 + 10.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + (-6.07 - 10.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.12 + 3.67i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.55816461903474801452071174415, −9.716000651840811855701971492883, −9.069608173495095238280091057033, −7.58438582964142916626900863186, −6.64966849559439760776479872930, −6.16622587011179917627777449170, −4.89504486389103755638849636862, −3.53277522930080434201623742906, −2.44822247915707954329730054155, −0.73766303773464805629608374416,
1.02898554970617843243155284339, 3.23108043251551330106186609493, 4.34892631008630096576797528912, 5.51177763020391113198840651295, 6.26555253398945424632950616018, 7.24528545487243420708809945579, 8.146327791415574641993918689316, 8.804429149694451368728928703574, 9.628589538790778293847125263557, 10.95775110827630745662203310536