| L(s) = 1 | − i·5-s + 6i·13-s + 2i·19-s − 6·23-s − 25-s − 6i·29-s + 6i·37-s − 6·41-s − 8i·43-s − 6·47-s − 7·49-s + 6i·53-s − 12i·59-s + 12i·61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 1.66i·13-s + 0.458i·19-s − 1.25·23-s − 0.200·25-s − 1.11i·29-s + 0.986i·37-s − 0.937·41-s − 1.21i·43-s − 0.875·47-s − 49-s + 0.824i·53-s − 1.56i·59-s + 1.53i·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6546362945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6546362945\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.024515545446805788796255488814, −8.338065784495452584938637779904, −7.64554800839939412890436895727, −6.66288772773669141880304451924, −6.13286969955302874772026971433, −5.11453701780389351011644204302, −4.31866496076204416168585177114, −3.66007602682121687698846397512, −2.29520985779362769365161074435, −1.47276750539579085937816044710,
0.19923098264716815049171522047, 1.69588556874723385494053368012, 2.90135097369070210518995871444, 3.48142902360063806680033416093, 4.63287132984373017747901413148, 5.46342621931611544457078932591, 6.17556414291540811859590672395, 7.00396912561056254421911877706, 7.83099967427201547758623682739, 8.312549474109581700981712043919
