| L(s) = 1 | − 2·2-s + 4·4-s + 5i·5-s − 20.6·7-s − 8·8-s − 10i·10-s + 11.4i·11-s + 15.3i·13-s + 41.3·14-s + 16·16-s − 99.5i·17-s + (76.3 − 32.0i)19-s + 20i·20-s − 22.8i·22-s + 77.3i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447i·5-s − 1.11·7-s − 0.353·8-s − 0.316i·10-s + 0.313i·11-s + 0.327i·13-s + 0.789·14-s + 0.250·16-s − 1.42i·17-s + (0.922 − 0.387i)19-s + 0.223i·20-s − 0.221i·22-s + 0.700i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2574615320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2574615320\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
| 19 | \( 1 + (-76.3 + 32.0i)T \) |
| good | 7 | \( 1 + 20.6T + 343T^{2} \) |
| 11 | \( 1 - 11.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 15.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 99.5iT - 4.91e3T^{2} \) |
| 23 | \( 1 - 77.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 48.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 70.6iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 271. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 278.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 0.799T + 7.95e4T^{2} \) |
| 47 | \( 1 - 615. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 458.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 246.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 498.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 622.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 428.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 306. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 64.1iT - 9.12e5T^{2} \) |
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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
−8.954431738353051648173009529814, −7.75107582303976337477095396324, −7.17729201792862791557799785715, −6.53725371105471039295119754130, −5.65529368608309211558054687099, −4.54865495294497863107387988707, −3.21091347492444023676949219552, −2.75065840582879862973041438775, −1.32905791976742713674801204057, −0.088165560263593705316330942130,
0.894420567903002771320843209883, 2.11688703035953578100374503170, 3.27398030259789065562554343538, 4.03024294722381858588702234552, 5.45739986510082270999081307549, 6.08183007161347455502410624864, 6.88801362020232740538153547596, 7.84503342756162781696062956830, 8.468774924099910705880682303782, 9.303143868838392488821858825576
