| 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
| \(L(1)\) |
\(\approx\) |
\(0.293958 + 0.709678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.293958 + 0.709678i\) |
| \(L(\frac{3}{2})\) |
|
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
−11.87155743444823576618691414129, −10.47480634983299212945983286650, −9.795043619028065311157014176521, −9.289120191022971870579262648242, −8.022973344092631875977933620594, −6.74300109421012252853259468529, −6.12222230875314374987170042074, −4.58660694737531538003861714253, −3.75572216193165549442230518239, −2.30795357271118856102465366246,
0.48679498189518907930183266173, 2.70932525750295076442157014770, 3.59263299891930224026818578911, 5.39213430206201839831945637149, 6.30498517475523438238710455379, 7.04387989340392764368635757257, 8.303111151110790689211383562790, 9.031461175966213677931290791137, 10.18122931162721848567067732934, 10.91923308779591399917835911238
