| L(s) = 1 | + (10 + 5i)5-s + 10i·7-s + 46·11-s + 34i·13-s − 66i·17-s − 104·19-s + 164i·23-s + (75 + 100i)25-s + 224·29-s − 72·31-s + (−50 + 100i)35-s − 22i·37-s − 194·41-s − 108i·43-s + 480i·47-s + ⋯ |
| L(s) = 1 | + (0.894 + 0.447i)5-s + 0.539i·7-s + 1.26·11-s + 0.725i·13-s − 0.941i·17-s − 1.25·19-s + 1.48i·23-s + (0.599 + 0.800i)25-s + 1.43·29-s − 0.417·31-s + (−0.241 + 0.482i)35-s − 0.0977i·37-s − 0.738·41-s − 0.383i·43-s + 1.48i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.232079766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.232079766\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-10 - 5i)T \) |
| good | 7 | \( 1 - 10iT - 343T^{2} \) |
| 11 | \( 1 - 46T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 104T + 6.85e3T^{2} \) |
| 23 | \( 1 - 164iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 224T + 2.43e4T^{2} \) |
| 31 | \( 1 + 72T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 194T + 6.89e4T^{2} \) |
| 43 | \( 1 + 108iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 480iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 286iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 426T + 2.05e5T^{2} \) |
| 61 | \( 1 - 698T + 2.26e5T^{2} \) |
| 67 | \( 1 - 328iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 188T + 3.57e5T^{2} \) |
| 73 | \( 1 - 740iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 412iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3iT - 9.12e5T^{2} \) |
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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
−11.29538145624414838549198783694, −10.12871929808344466932678234674, −9.313642687639383407151901736328, −8.684455406410368199689175610371, −7.11514844486315936926220172683, −6.42409931247853907755329678899, −5.44267021980031410157725107356, −4.11945471170727165535028091224, −2.68614745937563140591840769981, −1.49789329027816537129462274198,
0.834383004547964715258259548850, 2.15379328365932639722440774061, 3.81778435177005600299055533724, 4.85157262227652770006547910302, 6.17964246343205437727121460810, 6.74261670605340473633676975745, 8.345636407036975257591176369287, 8.852793287731828293621754148733, 10.19301282663833798436941161575, 10.49143788510375611610392382679
