L(s) = 1 | + (−0.923 + 1.60i)2-s + (−0.5 − 0.866i)3-s + (−1.20 − 2.09i)4-s + (0.382 − 0.662i)5-s + 1.84·6-s + 2.61·8-s + (−0.499 + 0.866i)9-s + (0.707 + 1.22i)10-s + (0.923 + 1.60i)11-s + (−1.20 + 2.09i)12-s − 13-s − 0.765·15-s + (−1.20 + 2.09i)16-s + (−0.923 − 1.60i)18-s − 1.84·20-s + ⋯ |
L(s) = 1 | + (−0.923 + 1.60i)2-s + (−0.5 − 0.866i)3-s + (−1.20 − 2.09i)4-s + (0.382 − 0.662i)5-s + 1.84·6-s + 2.61·8-s + (−0.499 + 0.866i)9-s + (0.707 + 1.22i)10-s + (0.923 + 1.60i)11-s + (−1.20 + 2.09i)12-s − 13-s − 0.765·15-s + (−1.20 + 2.09i)16-s + (−0.923 − 1.60i)18-s − 1.84·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5524889482\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5524889482\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 0.765T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 0.765T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - 1.84T + T^{2} \) |
| 89 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.415207071886573406393934987956, −8.680848409828330737293405755658, −7.75353019627381015154804703171, −7.22241996469806786467004795754, −6.68408803780852778874254812385, −5.86143835199875297170447651456, −5.06134951675157196760628294721, −4.50176805905612089524876100345, −2.09518598820693393259355413120, −1.11076249765841372462635910664,
0.73179004147242198548520352878, 2.29521423180542013200533193948, 3.22116140259266537153710783466, 3.79875805411527859739085903414, 4.85755724694160315751770063013, 5.98977845178304418029537669471, 6.87566564577993467265032906115, 8.154656968030285805445040387504, 8.802730297853633397307169627209, 9.565739629424013785584287272319