| L(s) = 1 | − 5i·3-s + 2i·7-s + 2·9-s − 39·11-s + 84i·13-s + 61i·17-s + 151·19-s + 10·21-s + 58i·23-s − 145i·27-s − 192·29-s + 18·31-s + 195i·33-s + 138i·37-s + 420·39-s + ⋯ |
| L(s) = 1 | − 0.962i·3-s + 0.107i·7-s + 0.0740·9-s − 1.06·11-s + 1.79i·13-s + 0.870i·17-s + 1.82·19-s + 0.103·21-s + 0.525i·23-s − 1.03i·27-s − 1.22·29-s + 0.104·31-s + 1.02i·33-s + 0.613i·37-s + 1.72·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.641647825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.641647825\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5iT - 27T^{2} \) |
| 7 | \( 1 - 2iT - 343T^{2} \) |
| 11 | \( 1 + 39T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 61iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 151T + 6.85e3T^{2} \) |
| 23 | \( 1 - 58iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 192T + 2.43e4T^{2} \) |
| 31 | \( 1 - 18T + 2.97e4T^{2} \) |
| 37 | \( 1 - 138iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 229T + 6.89e4T^{2} \) |
| 43 | \( 1 - 164iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 212iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 578iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 336T + 2.05e5T^{2} \) |
| 61 | \( 1 - 858T + 2.26e5T^{2} \) |
| 67 | \( 1 + 209iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 780T + 3.57e5T^{2} \) |
| 73 | \( 1 + 403iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 230T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 382iT - 9.12e5T^{2} \) |
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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
−11.12465080072562340770867535757, −9.918054300281964762245625564133, −9.103346589423949317988264741806, −7.85133730311744519298889962255, −7.29851259318978972731056174477, −6.30567275052962288945521705611, −5.21753486656324671773540152447, −3.88474608032378130037057038908, −2.33795150742758746190031377900, −1.26339501294110320429213289339,
0.60961234309670321078894777049, 2.73829305316966324934325726136, 3.70185621237209127973616967117, 5.12173810887201299868240399204, 5.50418318458000514619121336293, 7.28530267314271904419522932127, 7.910282377596835729813208521935, 9.209445740719670952106610329777, 9.990878595612598660703277764377, 10.58128241432074999638222300701
