| L(s) = 1 | − 0.585·3-s + 5-s − 2.65·9-s − 4.82·11-s + 0.828·13-s − 0.585·15-s + 5.41·17-s − 3.41·19-s + 6.82·23-s + 25-s + 3.31·27-s + 0.828·29-s − 2.82·31-s + 2.82·33-s + 3.65·37-s − 0.485·39-s + 11.0·41-s + 3.17·43-s − 2.65·45-s − 10.8·47-s − 3.17·51-s − 10.4·53-s − 4.82·55-s + 2·57-s − 11.4·59-s − 13.3·61-s + 0.828·65-s + ⋯ |
| L(s) = 1 | − 0.338·3-s + 0.447·5-s − 0.885·9-s − 1.45·11-s + 0.229·13-s − 0.151·15-s + 1.31·17-s − 0.783·19-s + 1.42·23-s + 0.200·25-s + 0.637·27-s + 0.153·29-s − 0.508·31-s + 0.492·33-s + 0.601·37-s − 0.0777·39-s + 1.72·41-s + 0.483·43-s − 0.396·45-s − 1.57·47-s − 0.444·51-s − 1.44·53-s − 0.651·55-s + 0.264·57-s − 1.48·59-s − 1.70·61-s + 0.102·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.585T + 3T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 - 1.17T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−8.004244852742722880106199917132, −7.53523454045351451531132927466, −6.43660811306357188989470610103, −5.79959246322817348060878706750, −5.24980156820310574439749172021, −4.49212750388630077394297344160, −3.09705660914852860856788341149, −2.71518107405165525246788488776, −1.35597280925506657523864388447, 0,
1.35597280925506657523864388447, 2.71518107405165525246788488776, 3.09705660914852860856788341149, 4.49212750388630077394297344160, 5.24980156820310574439749172021, 5.79959246322817348060878706750, 6.43660811306357188989470610103, 7.53523454045351451531132927466, 8.004244852742722880106199917132
