| L(s) = 1 | − 1.73i·3-s + 3.46·7-s − 2.99·9-s + 3.46·11-s + 4i·13-s + 6·17-s + 3.46i·19-s − 5.99i·21-s + 3.46i·23-s + 5.19i·27-s − 6i·29-s + 3.46i·31-s − 5.99i·33-s + 4i·37-s + 6.92·39-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + 1.30·7-s − 0.999·9-s + 1.04·11-s + 1.10i·13-s + 1.45·17-s + 0.794i·19-s − 1.30i·21-s + 0.722i·23-s + 0.999i·27-s − 1.11i·29-s + 0.622i·31-s − 1.04i·33-s + 0.657i·37-s + 1.10·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.027471256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.027471256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 12iT - 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.473586194239134055903077680774, −8.754278322040558262300179451619, −7.83967256919149816289581386646, −7.41379142499284469478699305407, −6.33512694048389246138360723196, −5.61958922686158489587338249905, −4.52109332751503868758612936892, −3.44123645151480479574393018605, −1.90991962484869984031872363737, −1.28130334964416257734969776934,
1.13398123254098601259281455735, 2.76202907071169921014993198063, 3.77005920092630376544209557033, 4.74217799323995507355433256998, 5.35291526840958205757561686416, 6.30500233653912990937573092000, 7.63967031131415679603442135426, 8.226761309196848248916706979656, 9.085056188758637943180008731287, 9.795055741033668147004457756477
