| L(s) = 1 | + 16·5-s + 7·7-s + 48·11-s − 39·13-s + 11·17-s − 74·19-s + 25·23-s + 131·25-s + 265·29-s + 91·31-s + 112·35-s − 54·37-s − 22·41-s + 173·43-s + 310·47-s + 49·49-s − 113·53-s + 768·55-s + 639·59-s + 478·61-s − 624·65-s − 831·67-s − 729·71-s + 178·73-s + 336·77-s − 1.04e3·79-s + 60·83-s + ⋯ |
| L(s) = 1 | + 1.43·5-s + 0.377·7-s + 1.31·11-s − 0.832·13-s + 0.156·17-s − 0.893·19-s + 0.226·23-s + 1.04·25-s + 1.69·29-s + 0.527·31-s + 0.540·35-s − 0.239·37-s − 0.0838·41-s + 0.613·43-s + 0.962·47-s + 1/7·49-s − 0.292·53-s + 1.88·55-s + 1.41·59-s + 1.00·61-s − 1.19·65-s − 1.51·67-s − 1.21·71-s + 0.285·73-s + 0.497·77-s − 1.48·79-s + 0.0793·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.437690227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.437690227\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 11 T + p^{3} T^{2} \) |
| 19 | \( 1 + 74 T + p^{3} T^{2} \) |
| 23 | \( 1 - 25 T + p^{3} T^{2} \) |
| 29 | \( 1 - 265 T + p^{3} T^{2} \) |
| 31 | \( 1 - 91 T + p^{3} T^{2} \) |
| 37 | \( 1 + 54 T + p^{3} T^{2} \) |
| 41 | \( 1 + 22 T + p^{3} T^{2} \) |
| 43 | \( 1 - 173 T + p^{3} T^{2} \) |
| 47 | \( 1 - 310 T + p^{3} T^{2} \) |
| 53 | \( 1 + 113 T + p^{3} T^{2} \) |
| 59 | \( 1 - 639 T + p^{3} T^{2} \) |
| 61 | \( 1 - 478 T + p^{3} T^{2} \) |
| 67 | \( 1 + 831 T + p^{3} T^{2} \) |
| 71 | \( 1 + 729 T + p^{3} T^{2} \) |
| 73 | \( 1 - 178 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1046 T + p^{3} T^{2} \) |
| 83 | \( 1 - 60 T + p^{3} T^{2} \) |
| 89 | \( 1 - 191 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1576 T + p^{3} T^{2} \) |
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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
−9.041798997306682493850571411478, −8.609829052132717106228665205190, −7.35897696013587178569893428986, −6.53046334075772985084845428369, −5.95510951454772306208792623838, −4.96799733251776559330260614703, −4.17782205190640191895510737858, −2.77780685727898795484093378126, −1.92526768428735194844152509660, −0.956015491852548421172638978694,
0.956015491852548421172638978694, 1.92526768428735194844152509660, 2.77780685727898795484093378126, 4.17782205190640191895510737858, 4.96799733251776559330260614703, 5.95510951454772306208792623838, 6.53046334075772985084845428369, 7.35897696013587178569893428986, 8.609829052132717106228665205190, 9.041798997306682493850571411478
