| L(s) = 1 | + i·3-s + (−1 − 2i)5-s + i·7-s − 9-s + 2·11-s + 2i·13-s + (2 − i)15-s − 8i·17-s − 2·19-s − 21-s + (−3 + 4i)25-s − i·27-s + 6·29-s − 6·31-s + 2i·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.447 − 0.894i)5-s + 0.377i·7-s − 0.333·9-s + 0.603·11-s + 0.554i·13-s + (0.516 − 0.258i)15-s − 1.94i·17-s − 0.458·19-s − 0.218·21-s + (−0.600 + 0.800i)25-s − 0.192i·27-s + 1.11·29-s − 1.07·31-s + 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.260578361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.260578361\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 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
−9.192729664935726728486477390928, −8.706378588763541812595149100941, −7.68348367633882874211988838190, −6.87958100851501840619047971212, −5.78161833934511795517188576961, −4.94792711066659320369407554227, −4.32603589485085588157140105334, −3.36994224273942612289903640158, −2.10433332557334575626803533222, −0.52566280255827036314115637280,
1.24768106070456405160249441195, 2.50892694688275645692797230169, 3.57095466389320635486728011979, 4.26572485536915366952131009732, 5.67259094331321193561409582100, 6.50385811052336219938951211551, 6.93501376367607572373007075236, 8.128300178415464168042752227638, 8.237814223888623306962877127107, 9.566878562646041423042272257843
