| L(s) = 1 | − i·3-s − 2i·5-s + 7-s − 9-s − 2·15-s − 4i·19-s − i·21-s + 6·23-s + 25-s + i·27-s − 2i·29-s − 2i·35-s − 2i·37-s + 4·41-s − 2i·43-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.894i·5-s + 0.377·7-s − 0.333·9-s − 0.516·15-s − 0.917i·19-s − 0.218i·21-s + 1.25·23-s + 0.200·25-s + 0.192i·27-s − 0.371i·29-s − 0.338i·35-s − 0.328i·37-s + 0.624·41-s − 0.304i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.790559636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.790559636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.000353381878619036790957916009, −7.16696083748123591290599086818, −6.62211916352716495129799527957, −5.64741630789959615869042758384, −4.99749876201679971901622117659, −4.42207316066328367752068191857, −3.28838660108542139215064469015, −2.36934680621902340384130475777, −1.37030158295408208278010045528, −0.51388270376163982729963428991,
1.22962484513991110304231013265, 2.45176894348710131164619845479, 3.19779946530125404744260478560, 3.91714185194938434222110712223, 4.84345097603506664326813463747, 5.45898724049720995372233139718, 6.38759074151353713177549970233, 6.93937441504709718051577003506, 7.78439896490761952128379440382, 8.395741912001371956819628310906
