| L(s) = 1 | − 4.02·5-s + 4.61·7-s − 1.53·11-s − 5.57·13-s + 1.29·17-s + 4.35·19-s − 4·23-s + 11.2·25-s + 1.86·29-s − 2.77·31-s − 18.5·35-s + 3.97·37-s + 3.03·41-s + 3.03·43-s − 9.65·47-s + 14.2·49-s + 8.58·53-s + 6.16·55-s − 5.73·59-s − 7.64·61-s + 22.4·65-s − 8.79·67-s + 6.88·71-s − 3.50·73-s − 7.06·77-s + 8.18·79-s − 4.12·83-s + ⋯ |
| L(s) = 1 | − 1.80·5-s + 1.74·7-s − 0.461·11-s − 1.54·13-s + 0.314·17-s + 1.00·19-s − 0.834·23-s + 2.24·25-s + 0.345·29-s − 0.498·31-s − 3.14·35-s + 0.653·37-s + 0.473·41-s + 0.462·43-s − 1.40·47-s + 2.04·49-s + 1.17·53-s + 0.831·55-s − 0.746·59-s − 0.979·61-s + 2.78·65-s − 1.07·67-s + 0.816·71-s − 0.409·73-s − 0.804·77-s + 0.920·79-s − 0.452·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.02T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 + 5.57T + 13T^{2} \) |
| 17 | \( 1 - 1.29T + 17T^{2} \) |
| 19 | \( 1 - 4.35T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 - 3.97T + 37T^{2} \) |
| 41 | \( 1 - 3.03T + 41T^{2} \) |
| 43 | \( 1 - 3.03T + 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 - 8.58T + 53T^{2} \) |
| 59 | \( 1 + 5.73T + 59T^{2} \) |
| 61 | \( 1 + 7.64T + 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 - 6.88T + 71T^{2} \) |
| 73 | \( 1 + 3.50T + 73T^{2} \) |
| 79 | \( 1 - 8.18T + 79T^{2} \) |
| 83 | \( 1 + 4.12T + 83T^{2} \) |
| 89 | \( 1 + 7.98T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−7.44825597006832592645096588132, −7.22119981124803573074481193629, −5.86350100999126382601908699032, −4.92753807539620679524477719051, −4.73546622840489348481698698930, −3.98339193367130843221693356100, −3.09561240874680195913430193185, −2.23073696435743740634222633343, −1.10320239994262060691076389474, 0,
1.10320239994262060691076389474, 2.23073696435743740634222633343, 3.09561240874680195913430193185, 3.98339193367130843221693356100, 4.73546622840489348481698698930, 4.92753807539620679524477719051, 5.86350100999126382601908699032, 7.22119981124803573074481193629, 7.44825597006832592645096588132
