| L(s) = 1 | − 5-s + 7-s + 11-s + 13-s − 2·17-s + 19-s + 4·23-s − 4·25-s + 5·29-s − 4·31-s − 35-s + 3·37-s + 6·41-s − 2·43-s − 9·47-s + 49-s − 2·53-s − 55-s + 59-s + 2·61-s − 65-s + 11·67-s − 2·71-s − 11·73-s + 77-s − 14·79-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.301·11-s + 0.277·13-s − 0.485·17-s + 0.229·19-s + 0.834·23-s − 4/5·25-s + 0.928·29-s − 0.718·31-s − 0.169·35-s + 0.493·37-s + 0.937·41-s − 0.304·43-s − 1.31·47-s + 1/7·49-s − 0.274·53-s − 0.134·55-s + 0.130·59-s + 0.256·61-s − 0.124·65-s + 1.34·67-s − 0.237·71-s − 1.28·73-s + 0.113·77-s − 1.57·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.86809494890915, −14.39109838087583, −14.04042882112827, −13.22112054999101, −12.97440063997482, −12.32395070185500, −11.67381932940094, −11.30179314788875, −10.98078470854316, −10.17300057378212, −9.708127759490657, −9.092102609910728, −8.486817806174441, −8.155226261574250, −7.363115398622364, −7.041560603935792, −6.276003287936231, −5.769969547512541, −5.021008684960288, −4.474612868842615, −3.924416022266254, −3.231415486740306, −2.562914350845758, −1.705475948189297, −1.010704607516215, 0,
1.010704607516215, 1.705475948189297, 2.562914350845758, 3.231415486740306, 3.924416022266254, 4.474612868842615, 5.021008684960288, 5.769969547512541, 6.276003287936231, 7.041560603935792, 7.363115398622364, 8.155226261574250, 8.486817806174441, 9.092102609910728, 9.708127759490657, 10.17300057378212, 10.98078470854316, 11.30179314788875, 11.67381932940094, 12.32395070185500, 12.97440063997482, 13.22112054999101, 14.04042882112827, 14.39109838087583, 14.86809494890915
