| L(s) = 1 | + 1.30·3-s + 2.30·5-s − 1.30·9-s − 6.60·13-s + 3·15-s − 17-s − 6.60·19-s + 0.302·25-s − 5.60·27-s − 4.30·31-s − 2.60·37-s − 8.60·39-s − 3.90·41-s + 7.30·43-s − 3.00·45-s − 4.60·47-s − 1.30·51-s − 3.69·53-s − 8.60·57-s − 9.21·59-s + 7.90·61-s − 15.2·65-s − 1.69·67-s − 7.81·71-s + 7.90·73-s + 0.394·75-s + 12.6·79-s + ⋯ |
| L(s) = 1 | + 0.752·3-s + 1.02·5-s − 0.434·9-s − 1.83·13-s + 0.774·15-s − 0.242·17-s − 1.51·19-s + 0.0605·25-s − 1.07·27-s − 0.772·31-s − 0.428·37-s − 1.37·39-s − 0.610·41-s + 1.11·43-s − 0.447·45-s − 0.671·47-s − 0.182·51-s − 0.507·53-s − 1.13·57-s − 1.19·59-s + 1.01·61-s − 1.88·65-s − 0.207·67-s − 0.927·71-s + 0.925·73-s + 0.0455·75-s + 1.41·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 19 | \( 1 + 6.60T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 41 | \( 1 + 3.90T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 + 9.21T + 59T^{2} \) |
| 61 | \( 1 - 7.90T + 61T^{2} \) |
| 67 | \( 1 + 1.69T + 67T^{2} \) |
| 71 | \( 1 + 7.81T + 71T^{2} \) |
| 73 | \( 1 - 7.90T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 6.30T + 97T^{2} \) |
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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
−8.310230596592226970569694608888, −7.60429034101457519171813928042, −6.77275661857174000064795688344, −5.99304240476876300710335641998, −5.21708031830030720063256730280, −4.42010476337939871781300369976, −3.30186685727323101955848324798, −2.32159944497486696868374542878, −1.98685144373014751336711921079, 0,
1.98685144373014751336711921079, 2.32159944497486696868374542878, 3.30186685727323101955848324798, 4.42010476337939871781300369976, 5.21708031830030720063256730280, 5.99304240476876300710335641998, 6.77275661857174000064795688344, 7.60429034101457519171813928042, 8.310230596592226970569694608888
