| L(s) = 1 | − 2i·5-s − 1.41i·7-s + 2.82·11-s + 4.24i·17-s + 6·23-s + 25-s − 2i·29-s + 1.41i·31-s − 2.82·35-s + 8.48·37-s + 4.24i·41-s + 12i·43-s − 6·47-s + 5·49-s − 2i·53-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s − 0.534i·7-s + 0.852·11-s + 1.02i·17-s + 1.25·23-s + 0.200·25-s − 0.371i·29-s + 0.254i·31-s − 0.478·35-s + 1.39·37-s + 0.662i·41-s + 1.82i·43-s − 0.875·47-s + 0.714·49-s − 0.274i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.172071351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.172071351\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 1.41iT - 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 7.07iT - 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.269526608609768875934796099969, −7.65030707926849508613728973125, −6.67615870592554962841218116084, −6.19348239192957817215293185603, −5.16305071623802989616731269182, −4.50178476061220472653184600969, −3.85341110080769410874840022545, −2.86877197789763021063941335532, −1.53656896057449347115426749338, −0.860425784347359018147467325817,
0.880326705295356679750619668851, 2.22353594273255874016808925469, 2.93537198380924467361148719401, 3.72294713810704962596301106435, 4.71098097165369237702396536436, 5.51012040187427945865549096212, 6.28636059667145877813156478116, 7.08414439696727669920351200934, 7.33708467860894300488325978696, 8.591258587314931205922571490844
