| L(s) = 1 | − 0.732·3-s − 0.267·5-s − 0.732·7-s − 2.46·9-s − 2.46·13-s + 0.196·15-s + 5.73·17-s + 6.73·19-s + 0.535·21-s + 8.19·23-s − 4.92·25-s + 4·27-s + 2.46·29-s − 1.26·31-s + 0.196·35-s + 2.26·37-s + 1.80·39-s + 4.26·41-s + 8·43-s + 0.660·45-s − 6.73·47-s − 6.46·49-s − 4.19·51-s − 9.19·53-s − 4.92·57-s + 6.53·59-s + 13.4·61-s + ⋯ |
| L(s) = 1 | − 0.422·3-s − 0.119·5-s − 0.276·7-s − 0.821·9-s − 0.683·13-s + 0.0506·15-s + 1.39·17-s + 1.54·19-s + 0.116·21-s + 1.70·23-s − 0.985·25-s + 0.769·27-s + 0.457·29-s − 0.227·31-s + 0.0331·35-s + 0.372·37-s + 0.288·39-s + 0.666·41-s + 1.21·43-s + 0.0984·45-s − 0.981·47-s − 0.923·49-s − 0.587·51-s − 1.26·53-s − 0.652·57-s + 0.850·59-s + 1.72·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.206833129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206833129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 + 0.267T + 5T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 2.26T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 6.73T + 47T^{2} \) |
| 53 | \( 1 + 9.19T + 53T^{2} \) |
| 59 | \( 1 - 6.53T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 4.92T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 + 9.53T + 89T^{2} \) |
| 97 | \( 1 - 1.92T + 97T^{2} \) |
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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
−9.802161550819546812585081001875, −9.467612089350491576845786230370, −8.226826830735980442449362485001, −7.52388038622047373748382378937, −6.59560705627879109904258588039, −5.51786961192473824491672578920, −5.05968621871419552660102392511, −3.55436523156847137094569377198, −2.72307679155258250382105711917, −0.891336829581885798334747559141,
0.891336829581885798334747559141, 2.72307679155258250382105711917, 3.55436523156847137094569377198, 5.05968621871419552660102392511, 5.51786961192473824491672578920, 6.59560705627879109904258588039, 7.52388038622047373748382378937, 8.226826830735980442449362485001, 9.467612089350491576845786230370, 9.802161550819546812585081001875
