| L(s) = 1 | − 0.0511·3-s − 0.342·5-s − 2.99·9-s − 2.07·11-s + 4.56·13-s + 0.0175·15-s − 3.03·17-s + 5.96·19-s − 0.364·23-s − 4.88·25-s + 0.306·27-s − 0.771·29-s − 1.14·31-s + 0.105·33-s + 8.56·37-s − 0.233·39-s − 7.12·41-s + 3.89·43-s + 1.02·45-s + 9.73·47-s + 0.155·51-s + 3.62·53-s + 0.708·55-s − 0.305·57-s − 13.4·59-s − 12.5·61-s − 1.56·65-s + ⋯ |
| L(s) = 1 | − 0.0295·3-s − 0.153·5-s − 0.999·9-s − 0.624·11-s + 1.26·13-s + 0.00452·15-s − 0.735·17-s + 1.36·19-s − 0.0760·23-s − 0.976·25-s + 0.0590·27-s − 0.143·29-s − 0.205·31-s + 0.0184·33-s + 1.40·37-s − 0.0374·39-s − 1.11·41-s + 0.593·43-s + 0.152·45-s + 1.41·47-s + 0.0217·51-s + 0.497·53-s + 0.0955·55-s − 0.0404·57-s − 1.75·59-s − 1.61·61-s − 0.194·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.0511T + 3T^{2} \) |
| 5 | \( 1 + 0.342T + 5T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 - 5.96T + 19T^{2} \) |
| 23 | \( 1 + 0.364T + 23T^{2} \) |
| 29 | \( 1 + 0.771T + 29T^{2} \) |
| 31 | \( 1 + 1.14T + 31T^{2} \) |
| 37 | \( 1 - 8.56T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 - 3.89T + 43T^{2} \) |
| 47 | \( 1 - 9.73T + 47T^{2} \) |
| 53 | \( 1 - 3.62T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 6.98T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 8.40T + 73T^{2} \) |
| 79 | \( 1 - 3.14T + 79T^{2} \) |
| 83 | \( 1 - 2.77T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 4.00T + 97T^{2} \) |
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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
−7.80145137149241511237525099976, −7.25242832252786804281019570958, −6.04090000876731383795543991884, −5.88984893139611970175417650756, −4.93679862170841735561735902139, −4.03306113853385906595110338496, −3.22625662380822497252418926754, −2.48915087601228680277231954505, −1.28675627754182362974753625273, 0,
1.28675627754182362974753625273, 2.48915087601228680277231954505, 3.22625662380822497252418926754, 4.03306113853385906595110338496, 4.93679862170841735561735902139, 5.88984893139611970175417650756, 6.04090000876731383795543991884, 7.25242832252786804281019570958, 7.80145137149241511237525099976
