| L(s) = 1 | − 2.80·3-s − 2.80·5-s − 1.80·7-s + 4.85·9-s + 6.40·11-s + 7.85·15-s − 5.63·17-s + 1.66·19-s + 5.04·21-s + 3.09·23-s + 2.85·25-s − 5.18·27-s + 6.07·29-s + 3.58·31-s − 17.9·33-s + 5.04·35-s + 5.14·37-s − 7.09·41-s − 2.84·43-s − 13.5·45-s − 1.51·47-s − 3.75·49-s + 15.7·51-s − 10.7·53-s − 17.9·55-s − 4.66·57-s + 9.27·59-s + ⋯ |
| L(s) = 1 | − 1.61·3-s − 1.25·5-s − 0.681·7-s + 1.61·9-s + 1.93·11-s + 2.02·15-s − 1.36·17-s + 0.381·19-s + 1.10·21-s + 0.645·23-s + 0.570·25-s − 0.998·27-s + 1.12·29-s + 0.643·31-s − 3.12·33-s + 0.853·35-s + 0.846·37-s − 1.10·41-s − 0.433·43-s − 2.02·45-s − 0.221·47-s − 0.536·49-s + 2.20·51-s − 1.47·53-s − 2.42·55-s − 0.617·57-s + 1.20·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 2.80T + 5T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 - 6.40T + 11T^{2} \) |
| 17 | \( 1 + 5.63T + 17T^{2} \) |
| 19 | \( 1 - 1.66T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 - 6.07T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 - 5.14T + 37T^{2} \) |
| 41 | \( 1 + 7.09T + 41T^{2} \) |
| 43 | \( 1 + 2.84T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 9.27T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 - 2.17T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 1.95T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 0.0392T + 83T^{2} \) |
| 89 | \( 1 - 7.44T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−9.275952741475891056920891903989, −8.430566786689264417471350270128, −7.20323564416092729698219454844, −6.57443882390213890399533557480, −6.19034767338444332678553189935, −4.76922855397319758776846889593, −4.29183585262675965199133541329, −3.27552095824965796442407836309, −1.21049810161219143969889601002, 0,
1.21049810161219143969889601002, 3.27552095824965796442407836309, 4.29183585262675965199133541329, 4.76922855397319758776846889593, 6.19034767338444332678553189935, 6.57443882390213890399533557480, 7.20323564416092729698219454844, 8.430566786689264417471350270128, 9.275952741475891056920891903989
