| L(s) = 1 | − i·3-s + 2i·5-s − 7-s − 9-s + 4i·11-s − 2i·13-s + 2·15-s − 6·17-s − 4i·19-s + i·21-s + 25-s + i·27-s − 2i·29-s + 4·33-s − 2i·35-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.894i·5-s − 0.377·7-s − 0.333·9-s + 1.20i·11-s − 0.554i·13-s + 0.516·15-s − 1.45·17-s − 0.917i·19-s + 0.218i·21-s + 0.200·25-s + 0.192i·27-s − 0.371i·29-s + 0.696·33-s − 0.338i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.351273832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351273832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−7.905020969208708830589515573637, −7.15483660999884910995926633641, −6.82527471771493743365920261249, −6.19971877832963943881583946857, −5.19929833950075121981748658631, −4.41945821815079513859095439208, −3.45670132383912115271025607665, −2.51713108965538197495781262191, −2.04457705963608713460144089676, −0.47121682533533159680618815094,
0.78784547344530291759307318167, 1.98552356019295616254431454216, 3.12848301320029608046129462944, 3.83437554529041327994365089668, 4.66388071760170995845344542250, 5.20341813076062003026679397861, 6.20947951068514871153993092540, 6.54965393765185090164397829781, 7.76727096864747691293533377903, 8.455515470664661825353012362484
