| L(s) = 1 | − 19.3·5-s − 4.84·7-s − 61.0·11-s + 13·13-s + 41.7·17-s + 107.·19-s + 28.5·23-s + 249.·25-s + 89.8·29-s − 183.·31-s + 93.6·35-s + 418.·37-s + 142.·41-s + 71.0·43-s + 323.·47-s − 319.·49-s + 25.1·53-s + 1.18e3·55-s − 684.·59-s + 308.·61-s − 251.·65-s − 672.·67-s − 326.·71-s + 24.3·73-s + 295.·77-s − 166.·79-s − 201.·83-s + ⋯ |
| L(s) = 1 | − 1.72·5-s − 0.261·7-s − 1.67·11-s + 0.277·13-s + 0.596·17-s + 1.29·19-s + 0.258·23-s + 1.99·25-s + 0.575·29-s − 1.06·31-s + 0.452·35-s + 1.85·37-s + 0.543·41-s + 0.252·43-s + 1.00·47-s − 0.931·49-s + 0.0650·53-s + 2.89·55-s − 1.51·59-s + 0.646·61-s − 0.479·65-s − 1.22·67-s − 0.546·71-s + 0.0389·73-s + 0.437·77-s − 0.237·79-s − 0.265·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 19.3T + 125T^{2} \) |
| 7 | \( 1 + 4.84T + 343T^{2} \) |
| 11 | \( 1 + 61.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 41.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 28.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 89.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 71.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 25.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 684.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 672.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 326.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 24.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 201.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 108.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−8.245736428002200750136047556038, −7.57547644245008626663418372191, −7.38208841751880378861147131901, −6.01279702449559026804490910220, −5.12714336593394908950690043799, −4.32047385002478398692844630940, −3.33745089719184438097433035452, −2.75545666297865441718368239673, −0.958062341644736310805027065430, 0,
0.958062341644736310805027065430, 2.75545666297865441718368239673, 3.33745089719184438097433035452, 4.32047385002478398692844630940, 5.12714336593394908950690043799, 6.01279702449559026804490910220, 7.38208841751880378861147131901, 7.57547644245008626663418372191, 8.245736428002200750136047556038
