L(s) = 1 | − 3.02·3-s + 2.92·5-s + 2.58i·7-s + 6.15·9-s + i·11-s − 1.48i·13-s − 8.84·15-s + 0.518·17-s + (2.44 − 3.60i)19-s − 7.83i·21-s + 0.564i·23-s + 3.54·25-s − 9.56·27-s − 7.78i·29-s + 3.27·31-s + ⋯ |
L(s) = 1 | − 1.74·3-s + 1.30·5-s + 0.978i·7-s + 2.05·9-s + 0.301i·11-s − 0.413i·13-s − 2.28·15-s + 0.125·17-s + (0.561 − 0.827i)19-s − 1.70i·21-s + 0.117i·23-s + 0.708·25-s − 1.84·27-s − 1.44i·29-s + 0.587·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.218211158\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.218211158\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - iT \) |
| 19 | \( 1 + (-2.44 + 3.60i)T \) |
good | 3 | \( 1 + 3.02T + 3T^{2} \) |
| 5 | \( 1 - 2.92T + 5T^{2} \) |
| 7 | \( 1 - 2.58iT - 7T^{2} \) |
| 13 | \( 1 + 1.48iT - 13T^{2} \) |
| 17 | \( 1 - 0.518T + 17T^{2} \) |
| 23 | \( 1 - 0.564iT - 23T^{2} \) |
| 29 | \( 1 + 7.78iT - 29T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 + 5.18iT - 37T^{2} \) |
| 41 | \( 1 + 1.05iT - 41T^{2} \) |
| 43 | \( 1 + 4.86iT - 43T^{2} \) |
| 47 | \( 1 + 13.1iT - 47T^{2} \) |
| 53 | \( 1 - 6.61iT - 53T^{2} \) |
| 59 | \( 1 + 1.16T + 59T^{2} \) |
| 61 | \( 1 - 4.69T + 61T^{2} \) |
| 67 | \( 1 + 1.34T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 - 1.60iT - 83T^{2} \) |
| 89 | \( 1 - 12.5iT - 89T^{2} \) |
| 97 | \( 1 - 8.44iT - 97T^{2} \) |
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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
−8.769928694807839099695190476472, −7.55811336264638614311838962642, −6.76139113232382423123290010369, −6.06201322377680828305704691756, −5.54645555556331081859226539060, −5.18319459875799137919403494158, −4.19398054395861405871543228968, −2.66440041749012974938145271005, −1.81029555324968343648508555843, −0.57541763111581775635202594870,
1.00396479163129936646689875389, 1.63228075221475393983826288387, 3.19838294829741584735352073615, 4.40533119863637568118359096435, 4.98497048158511454627646187295, 5.85930341341532698663966472354, 6.21292249542772131272811800949, 6.95718988625498591826663010195, 7.65040673469907591004109234437, 8.855865161169004175336003833441