L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 3.44·5-s − 0.999·8-s + (1.72 − 2.98i)10-s − 2·11-s + (−2.44 + 4.24i)13-s + (−0.5 + 0.866i)16-s + (−1 + 1.73i)17-s + (3.72 + 6.45i)19-s + (−1.72 − 2.98i)20-s + (−1 + 1.73i)22-s + 23-s + 6.89·25-s + (2.44 + 4.24i)26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.54·5-s − 0.353·8-s + (0.545 − 0.944i)10-s − 0.603·11-s + (−0.679 + 1.17i)13-s + (−0.125 + 0.216i)16-s + (−0.242 + 0.420i)17-s + (0.854 + 1.48i)19-s + (−0.385 − 0.667i)20-s + (−0.213 + 0.369i)22-s + 0.208·23-s + 1.37·25-s + (0.480 + 0.832i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.637457135\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.637457135\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.44T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (2.44 - 4.24i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.72 - 6.45i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + (-1.44 - 2.51i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.89 - 6.75i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.89 + 8.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.44 - 2.51i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.89 + 8.48i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.550 - 0.953i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.72 + 9.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.55 - 2.68i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 + (-1.44 + 2.51i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.94 - 6.84i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.55 - 6.14i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.44 + 5.97i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.119431100380605397754153668502, −8.297649622562028412962528477874, −7.15850980680840514532411217176, −6.39085839089398548989726409934, −5.60797095831921694116980187079, −5.05321508738326372343903427207, −4.07716418972361414240284614163, −2.92862015896377063784862479057, −2.10842928785853012845684540391, −1.35394396874967684758053898140,
0.78372313184061596955357122483, 2.57793306275771064184363887089, 2.75324408627812732753980893456, 4.41186245535013464796791299862, 5.18651390174391905444547937727, 5.69747995975721397881395821644, 6.39797428085207380519831860006, 7.35221128636932316650741265365, 7.85957003928350219373722051672, 8.998550475214449266099758705973