| L(s) = 1 | − i·3-s + 4·7-s − 9-s − 4i·11-s + 4i·13-s − 2·17-s − 4i·19-s − 4i·21-s + 8·23-s + 5·25-s + i·27-s − 8i·29-s − 4·31-s − 4·33-s + 4i·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.51·7-s − 0.333·9-s − 1.20i·11-s + 1.10i·13-s − 0.485·17-s − 0.917i·19-s − 0.872i·21-s + 1.66·23-s + 25-s + 0.192i·27-s − 1.48i·29-s − 0.718·31-s − 0.696·33-s + 0.657i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42600 - 0.590671i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42600 - 0.590671i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 8iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−11.41089745886956894689233562895, −10.65523739486629261803172297001, −8.953151142074650757931909502055, −8.607929664938068288413437336413, −7.44802858609941989247356452329, −6.59814620411722924341655163695, −5.34035296335155650752794073861, −4.39028236038812750669558500184, −2.69898805308456660117006150632, −1.26469043566247071908399034646,
1.71617938564524156237009286279, 3.35229872855306792621407134492, 4.89032926922343161752226705007, 5.11859523271144418185621408864, 6.83644889163487003112980183420, 7.85199416717843507728445762695, 8.653391827828614005941414101820, 9.678671396398343345240941652831, 10.76049071635102347525383789749, 11.10875095920167907921317979099
