| L(s) = 1 | + 3-s + 5.19i·5-s − 8·9-s − 17·11-s + 13.8i·13-s + 5.19i·15-s + 25·17-s − 7·19-s − 5.19i·23-s − 2·25-s − 17·27-s + 13.8i·29-s − 32.9i·31-s − 17·33-s + 8.66i·37-s + ⋯ |
| L(s) = 1 | + 0.333·3-s + 1.03i·5-s − 0.888·9-s − 1.54·11-s + 1.06i·13-s + 0.346i·15-s + 1.47·17-s − 0.368·19-s − 0.225i·23-s − 0.0800·25-s − 0.629·27-s + 0.477i·29-s − 1.06i·31-s − 0.515·33-s + 0.234i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1941158221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1941158221\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + 9T^{2} \) |
| 5 | \( 1 - 5.19iT - 25T^{2} \) |
| 11 | \( 1 + 17T + 121T^{2} \) |
| 13 | \( 1 - 13.8iT - 169T^{2} \) |
| 17 | \( 1 - 25T + 289T^{2} \) |
| 19 | \( 1 + 7T + 361T^{2} \) |
| 23 | \( 1 + 5.19iT - 529T^{2} \) |
| 29 | \( 1 - 13.8iT - 841T^{2} \) |
| 31 | \( 1 + 32.9iT - 961T^{2} \) |
| 37 | \( 1 - 8.66iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 26T + 1.68e3T^{2} \) |
| 43 | \( 1 + 14T + 1.84e3T^{2} \) |
| 47 | \( 1 + 50.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 91.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 55T + 3.48e3T^{2} \) |
| 61 | \( 1 + 22.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 17T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 119T + 5.32e3T^{2} \) |
| 79 | \( 1 + 74.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 110T + 6.88e3T^{2} \) |
| 89 | \( 1 + 71T + 7.92e3T^{2} \) |
| 97 | \( 1 - 22T + 9.40e3T^{2} \) |
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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
−8.896660513607987477015085825906, −8.105649483321005001398620506409, −7.47395661695115144518125509965, −6.59619170179851577151279267093, −5.73336071760605042147732584039, −4.90679852000016785958140336011, −3.55033285352623849854831467344, −2.88440552695958178421965357665, −2.01503952134575329597204720239, −0.05134611647929917832697418733,
1.16414907841412436686568804337, 2.66459453661010093686614488641, 3.31214947260280821509517195136, 4.71236773240013439816183100951, 5.42307752992470490630007123768, 5.91601775778785075510524616364, 7.48302095745222040265879324110, 8.032788605738437639023726569684, 8.557510280241137166173421812873, 9.403738542706125345880170624644
