| L(s) = 1 | + 0.732·5-s − 0.732·7-s + 1.46·11-s − 5.46·13-s − 0.732·17-s + 4.73·19-s − 23-s − 4.46·25-s − 5.46·29-s + 2.53·31-s − 0.535·35-s − 0.535·37-s + 6.92·41-s − 0.732·43-s − 7.46·47-s − 6.46·49-s − 14.1·53-s + 1.07·55-s + 0.535·59-s + 6.39·61-s − 4·65-s + 6.19·67-s − 10.9·71-s − 6·73-s − 1.07·77-s + 9.12·79-s − 16.3·83-s + ⋯ |
| L(s) = 1 | + 0.327·5-s − 0.276·7-s + 0.441·11-s − 1.51·13-s − 0.177·17-s + 1.08·19-s − 0.208·23-s − 0.892·25-s − 1.01·29-s + 0.455·31-s − 0.0905·35-s − 0.0881·37-s + 1.08·41-s − 0.111·43-s − 1.08·47-s − 0.923·49-s − 1.94·53-s + 0.144·55-s + 0.0697·59-s + 0.818·61-s − 0.496·65-s + 0.756·67-s − 1.29·71-s − 0.702·73-s − 0.122·77-s + 1.02·79-s − 1.79·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 0.732T + 43T^{2} \) |
| 47 | \( 1 + 7.46T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 - 0.535T + 59T^{2} \) |
| 61 | \( 1 - 6.39T + 61T^{2} \) |
| 67 | \( 1 - 6.19T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 6.19T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.126307258494187588757459562310, −7.52537286084822679858522638127, −6.80192346795087375928533669743, −5.98603092632582803814418416489, −5.22291143088075552363474369181, −4.44600130939613722601875113504, −3.44169964136410917302499369696, −2.54651845936113254931482402059, −1.54950818726962492418409304344, 0,
1.54950818726962492418409304344, 2.54651845936113254931482402059, 3.44169964136410917302499369696, 4.44600130939613722601875113504, 5.22291143088075552363474369181, 5.98603092632582803814418416489, 6.80192346795087375928533669743, 7.52537286084822679858522638127, 8.126307258494187588757459562310
