| L(s) = 1 | + (−0.873 + 2.05i)5-s + (−2.52 − 0.794i)7-s + 0.935i·11-s + 6.43·13-s − 0.439i·17-s + 3.43i·19-s − 8.69·23-s + (−3.47 − 3.59i)25-s + 3.38i·29-s + 6.60i·31-s + (3.83 − 4.50i)35-s − 9.91i·37-s + 3.89·41-s + 5.85i·43-s + 3.41i·47-s + ⋯ |
| L(s) = 1 | + (−0.390 + 0.920i)5-s + (−0.953 − 0.300i)7-s + 0.282i·11-s + 1.78·13-s − 0.106i·17-s + 0.788i·19-s − 1.81·23-s + (−0.695 − 0.718i)25-s + 0.628i·29-s + 1.18i·31-s + (0.648 − 0.760i)35-s − 1.62i·37-s + 0.607·41-s + 0.892i·43-s + 0.498i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4759721766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4759721766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.873 - 2.05i)T \) |
| 7 | \( 1 + (2.52 + 0.794i)T \) |
| good | 11 | \( 1 - 0.935iT - 11T^{2} \) |
| 13 | \( 1 - 6.43T + 13T^{2} \) |
| 17 | \( 1 + 0.439iT - 17T^{2} \) |
| 19 | \( 1 - 3.43iT - 19T^{2} \) |
| 23 | \( 1 + 8.69T + 23T^{2} \) |
| 29 | \( 1 - 3.38iT - 29T^{2} \) |
| 31 | \( 1 - 6.60iT - 31T^{2} \) |
| 37 | \( 1 + 9.91iT - 37T^{2} \) |
| 41 | \( 1 - 3.89T + 41T^{2} \) |
| 43 | \( 1 - 5.85iT - 43T^{2} \) |
| 47 | \( 1 - 3.41iT - 47T^{2} \) |
| 53 | \( 1 + 7.31T + 53T^{2} \) |
| 59 | \( 1 - 0.437T + 59T^{2} \) |
| 61 | \( 1 + 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 11.9iT - 67T^{2} \) |
| 71 | \( 1 - 7.68iT - 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 7.78T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−9.407273238961322471441334520818, −8.377037831676250611711544733223, −7.77866503793491268810857188442, −6.86739926874599071937188228524, −6.27477408366580127390530122667, −5.69098308977144809768662304725, −4.12919431109173084182255520422, −3.69128280197735515814884504239, −2.86265393298106032676169353184, −1.53746611861671348033172816252,
0.16266196700649775606268950194, 1.41208328492724184209940527547, 2.77112053731334756385080657082, 3.84541057178547557424803029106, 4.30217909625572144235297267466, 5.70603650322201937260363046015, 5.98917796390650766997981743788, 6.93987699485768608673224105987, 8.079853265591806852681156770304, 8.472466741495204716813566874578
