| L(s) = 1 | + i·3-s + 4.14i·5-s + (0.717 − 2.54i)7-s − 9-s + (0.170 + 3.31i)11-s − 3.09·13-s − 4.14·15-s − 3.57·17-s + 3.91·19-s + (2.54 + 0.717i)21-s − 7.71·23-s − 12.1·25-s − i·27-s + 1.65i·29-s − 9.23i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.85i·5-s + (0.271 − 0.962i)7-s − 0.333·9-s + (0.0514 + 0.998i)11-s − 0.857·13-s − 1.06·15-s − 0.867·17-s + 0.898·19-s + (0.555 + 0.156i)21-s − 1.60·23-s − 2.43·25-s − 0.192i·27-s + 0.308i·29-s − 1.65i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09324988905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09324988905\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.717 + 2.54i)T \) |
| 11 | \( 1 + (-0.170 - 3.31i)T \) |
| good | 5 | \( 1 - 4.14iT - 5T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 17 | \( 1 + 3.57T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 + 7.71T + 23T^{2} \) |
| 29 | \( 1 - 1.65iT - 29T^{2} \) |
| 31 | \( 1 + 9.23iT - 31T^{2} \) |
| 37 | \( 1 + 0.869T + 37T^{2} \) |
| 41 | \( 1 - 5.91T + 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 6.76iT - 47T^{2} \) |
| 53 | \( 1 + 1.88T + 53T^{2} \) |
| 59 | \( 1 - 10.1iT - 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 + 9.84T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 - 10.8iT - 79T^{2} \) |
| 83 | \( 1 + 5.67T + 83T^{2} \) |
| 89 | \( 1 - 10.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.52iT - 97T^{2} \) |
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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
−9.382383748950395998667643815148, −8.044292289234397762083221968444, −7.38732665293667849780394518942, −7.03853932682198050089429382372, −6.21208769980259806703643433418, −5.28693278100867989386020878919, −4.13524478640998431699458177325, −3.88003208201857213020341489481, −2.66495186334103703687095149479, −2.05357577666171707107321373283,
0.02682891758961204605574349308, 1.22332825681327055030204882739, 2.05921380796550509770012254482, 3.13064162566503620901274112637, 4.42862221674600560806981414304, 4.98613850937731876548906043103, 5.75498966101064950294497749447, 6.24855274884108562388455955709, 7.53227691789496238213693441662, 8.170605952229214307372372141440
