L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 − 2.59i)11-s + (0.499 − 0.866i)12-s + 4·13-s + (2 − 1.73i)14-s + (−0.5 + 0.866i)16-s + (−0.499 − 0.866i)18-s + (2 − 3.46i)19-s + (−0.500 − 2.59i)21-s + 3·22-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.408·6-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.452 − 0.783i)11-s + (0.144 − 0.249i)12-s + 1.10·13-s + (0.534 − 0.462i)14-s + (−0.125 + 0.216i)16-s + (−0.117 − 0.204i)18-s + (0.458 − 0.794i)19-s + (−0.109 − 0.566i)21-s + 0.639·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.239752778\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239752778\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - T + 97T^{2} \) |
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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
−9.744858627252230756232203674968, −9.101975859263661736758333345918, −8.376748417254120533624820626034, −7.54134283671819224331255027361, −6.49090782991453514098084755022, −5.88990624408301048799004922197, −4.77574269630849804260675490790, −3.69435296552545415382590779630, −2.78014600706319245005821475496, −0.74442779376075173490285700792,
1.13457140669304745613012047354, 2.46934182278326863449203643206, 3.29475552576276514913268018561, 4.37614593430037840563207157328, 5.75848231558810287171853896033, 6.57390257645950109053749176522, 7.52704049057369605583385235807, 8.364684890514208047091986603515, 9.062954095047115403321517442642, 9.973563051449953779557160817183