| L(s) = 1 | + (1.08 − 1.34i)3-s − 5-s + (−2.53 + 0.769i)7-s + (−0.641 − 2.93i)9-s + 1.53i·11-s + 1.09i·13-s + (−1.08 + 1.34i)15-s − 1.57·17-s + 4.32i·19-s + (−1.70 + 4.25i)21-s + 6.09i·23-s + 25-s + (−4.65 − 2.31i)27-s + 0.867i·29-s + 4.03i·31-s + ⋯ |
| L(s) = 1 | + (0.626 − 0.779i)3-s − 0.447·5-s + (−0.956 + 0.291i)7-s + (−0.213 − 0.976i)9-s + 0.462i·11-s + 0.302i·13-s + (−0.280 + 0.348i)15-s − 0.381·17-s + 0.992i·19-s + (−0.373 + 0.927i)21-s + 1.27i·23-s + 0.200·25-s + (−0.895 − 0.445i)27-s + 0.161i·29-s + 0.724i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.099590857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099590857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.08 + 1.34i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2.53 - 0.769i)T \) |
| good | 11 | \( 1 - 1.53iT - 11T^{2} \) |
| 13 | \( 1 - 1.09iT - 13T^{2} \) |
| 17 | \( 1 + 1.57T + 17T^{2} \) |
| 19 | \( 1 - 4.32iT - 19T^{2} \) |
| 23 | \( 1 - 6.09iT - 23T^{2} \) |
| 29 | \( 1 - 0.867iT - 29T^{2} \) |
| 31 | \( 1 - 4.03iT - 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 4.51iT - 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 - 10.7iT - 61T^{2} \) |
| 67 | \( 1 + 3.69T + 67T^{2} \) |
| 71 | \( 1 - 11.7iT - 71T^{2} \) |
| 73 | \( 1 + 2.71iT - 73T^{2} \) |
| 79 | \( 1 - 7.04T + 79T^{2} \) |
| 83 | \( 1 + 6.68T + 83T^{2} \) |
| 89 | \( 1 - 4.13T + 89T^{2} \) |
| 97 | \( 1 + 16.9iT - 97T^{2} \) |
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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
−9.437313084525242896435021503669, −8.689574445625346536633307207102, −7.83275089699117551830752767855, −7.20800827215645333090936582981, −6.42214525865989687462484474219, −5.67988376743223783906286400342, −4.27910534393873238165373737567, −3.42878960292492668987837602142, −2.56644682011090004064383387911, −1.35735736926566837878250479398,
0.39249533198076389114325171505, 2.49285837555507217971979394300, 3.19465135219185600492749539555, 4.13162943689102868319092615592, 4.80547074749733521802029412100, 6.01820889425487626259523683002, 6.80546033481323933707289657563, 7.83789744342599721062879071095, 8.421244826619210298908369028722, 9.374471214345540662095076810001
