| L(s) = 1 | − 1.53·5-s − 2.53·7-s + 4.41·11-s + 3.94·13-s − 3.69·17-s − 7.10·19-s + 4.71·23-s − 2.65·25-s − 3.87·29-s + 3.29·31-s + 3.87·35-s − 7.92·37-s + 5.33·41-s + 1.73·43-s + 0.467·47-s − 0.588·49-s + 6.80·53-s − 6.75·55-s − 8.04·59-s − 6.49·61-s − 6.04·65-s − 10.5·67-s − 7.46·71-s − 12.2·73-s − 11.1·77-s − 7.20·79-s + 10.6·83-s + ⋯ |
| L(s) = 1 | − 0.685·5-s − 0.957·7-s + 1.33·11-s + 1.09·13-s − 0.896·17-s − 1.63·19-s + 0.983·23-s − 0.530·25-s − 0.720·29-s + 0.591·31-s + 0.655·35-s − 1.30·37-s + 0.832·41-s + 0.264·43-s + 0.0682·47-s − 0.0840·49-s + 0.934·53-s − 0.911·55-s − 1.04·59-s − 0.830·61-s − 0.749·65-s − 1.29·67-s − 0.885·71-s − 1.43·73-s − 1.27·77-s − 0.810·79-s + 1.16·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 - 4.41T + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 + 3.69T + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 - 4.71T + 23T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 - 3.29T + 31T^{2} \) |
| 37 | \( 1 + 7.92T + 37T^{2} \) |
| 41 | \( 1 - 5.33T + 41T^{2} \) |
| 43 | \( 1 - 1.73T + 43T^{2} \) |
| 47 | \( 1 - 0.467T + 47T^{2} \) |
| 53 | \( 1 - 6.80T + 53T^{2} \) |
| 59 | \( 1 + 8.04T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 7.46T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 7.20T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 5.41T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−8.991494334685810374807034628198, −8.136204844803841439627193493822, −7.01696250776115332010469012673, −6.51043179914377592215632920998, −5.83464397209695134416620801429, −4.32576511966310388443167867858, −3.94130081581097275436103461484, −2.96097002112012774909468285680, −1.54824041768180344176173503980, 0,
1.54824041768180344176173503980, 2.96097002112012774909468285680, 3.94130081581097275436103461484, 4.32576511966310388443167867858, 5.83464397209695134416620801429, 6.51043179914377592215632920998, 7.01696250776115332010469012673, 8.136204844803841439627193493822, 8.991494334685810374807034628198
