| L(s) = 1 | + 2.61·3-s − 2.23·5-s − 2.38·7-s + 3.85·9-s − 4·11-s + 3.38·13-s − 5.85·15-s − 8.09·17-s − 8.47·19-s − 6.23·21-s + 0.854·23-s + 2.23·27-s − 1.76·29-s + 5.23·31-s − 10.4·33-s + 5.32·35-s + 9.94·37-s + 8.85·39-s + 2.23·41-s + 1.47·43-s − 8.61·45-s − 4.23·47-s − 1.32·49-s − 21.1·51-s − 4.76·53-s + 8.94·55-s − 22.1·57-s + ⋯ |
| L(s) = 1 | + 1.51·3-s − 0.999·5-s − 0.900·7-s + 1.28·9-s − 1.20·11-s + 0.937·13-s − 1.51·15-s − 1.96·17-s − 1.94·19-s − 1.36·21-s + 0.178·23-s + 0.430·27-s − 0.327·29-s + 0.940·31-s − 1.82·33-s + 0.900·35-s + 1.63·37-s + 1.41·39-s + 0.349·41-s + 0.224·43-s − 1.28·45-s − 0.617·47-s − 0.189·49-s − 2.96·51-s − 0.654·53-s + 1.20·55-s − 2.93·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 + 2.38T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 8.09T + 17T^{2} \) |
| 19 | \( 1 + 8.47T + 19T^{2} \) |
| 23 | \( 1 - 0.854T + 23T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 - 9.94T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 1.47T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 + 4.76T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 7.76T + 61T^{2} \) |
| 67 | \( 1 - 1.76T + 67T^{2} \) |
| 71 | \( 1 - 2.14T + 71T^{2} \) |
| 73 | \( 1 - 6.09T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.986334438992603557963508324155, −8.338615469316777067597171208370, −7.952529331634785272450076172199, −6.88303143350858411132248569214, −6.15620470186570284587004171258, −4.45956025348241075181913209410, −3.95347461120637317686712326336, −2.92042704929423559866580119039, −2.22552970308426052426653946758, 0,
2.22552970308426052426653946758, 2.92042704929423559866580119039, 3.95347461120637317686712326336, 4.45956025348241075181913209410, 6.15620470186570284587004171258, 6.88303143350858411132248569214, 7.952529331634785272450076172199, 8.338615469316777067597171208370, 8.986334438992603557963508324155
