| L(s) = 1 | + 3i·3-s + 8·4-s + 7i·7-s − 9·9-s + 42·11-s + 24i·12-s − 20i·13-s + 64·16-s + 66i·17-s − 38·19-s − 21·21-s − 12i·23-s − 27i·27-s + 56i·28-s + 258·29-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 4-s + 0.377i·7-s − 0.333·9-s + 1.15·11-s + 0.577i·12-s − 0.426i·13-s + 16-s + 0.941i·17-s − 0.458·19-s − 0.218·21-s − 0.108i·23-s − 0.192i·27-s + 0.377i·28-s + 1.65·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.696646993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.696646993\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
| good | 2 | \( 1 - 8T^{2} \) |
| 11 | \( 1 - 42T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 38T + 6.85e3T^{2} \) |
| 23 | \( 1 + 12iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 258T + 2.43e4T^{2} \) |
| 31 | \( 1 - 146T + 2.97e4T^{2} \) |
| 37 | \( 1 - 434iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 282T + 6.89e4T^{2} \) |
| 43 | \( 1 + 20iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 72iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 336iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 360T + 2.05e5T^{2} \) |
| 61 | \( 1 + 682T + 2.26e5T^{2} \) |
| 67 | \( 1 - 812iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 810T + 3.57e5T^{2} \) |
| 73 | \( 1 - 124iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 156iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.20e3iT - 9.12e5T^{2} \) |
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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
−10.43783698681441300205898460808, −10.03233418143989623071586665334, −8.716202870731758095154649048865, −8.105430557825315373819861707090, −6.64723828366536530061369390364, −6.23598897415082182123114994300, −4.95622995994598523781337071975, −3.73752777238727790701223310625, −2.69199363521054245406467646359, −1.33279706503025408355956237477,
0.895523348264301215479720250227, 2.04517768062009143156395693101, 3.22691249653683311225277698388, 4.53794232000824498554585916207, 5.99979181675769912948718610914, 6.73286216941067304053564033031, 7.33089677757983477518735664540, 8.394424547589801100640009632995, 9.405085659690589303865934197599, 10.44949032605941785146212110282
