| L(s) = 1 | − 7-s − 4·11-s + 13-s − 4·17-s + 19-s − 4·23-s + 4·29-s + 5·31-s + 6·37-s − 12·41-s + 5·43-s + 8·47-s − 6·49-s − 12·53-s − 8·59-s − 7·61-s + 13·67-s + 12·71-s − 6·73-s + 4·77-s − 12·79-s + 8·83-s − 91-s − 13·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s + 0.277·13-s − 0.970·17-s + 0.229·19-s − 0.834·23-s + 0.742·29-s + 0.898·31-s + 0.986·37-s − 1.87·41-s + 0.762·43-s + 1.16·47-s − 6/7·49-s − 1.64·53-s − 1.04·59-s − 0.896·61-s + 1.58·67-s + 1.42·71-s − 0.702·73-s + 0.455·77-s − 1.35·79-s + 0.878·83-s − 0.104·91-s − 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.212436788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212436788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.81225133366886, −15.68254803071956, −15.29255529711643, −14.21586462227242, −13.94907110058396, −13.22604682009453, −12.88299047359036, −12.19516422720310, −11.61856184859520, −10.89523607645291, −10.48742767756551, −9.815599196857762, −9.328305735901563, −8.461740338521804, −8.063851042548609, −7.444444890910200, −6.534590787026444, −6.235062758612995, −5.348210793766099, −4.725725280803287, −4.070931938052217, −3.112389082634009, −2.586087333749340, −1.689782126156918, −0.4696151680451682,
0.4696151680451682, 1.689782126156918, 2.586087333749340, 3.112389082634009, 4.070931938052217, 4.725725280803287, 5.348210793766099, 6.235062758612995, 6.534590787026444, 7.444444890910200, 8.063851042548609, 8.461740338521804, 9.328305735901563, 9.815599196857762, 10.48742767756551, 10.89523607645291, 11.61856184859520, 12.19516422720310, 12.88299047359036, 13.22604682009453, 13.94907110058396, 14.21586462227242, 15.29255529711643, 15.68254803071956, 15.81225133366886
