| L(s) = 1 | + 2-s + 4-s − 2.40·5-s + 2.01·7-s + 8-s − 2.40·10-s − 4.74·11-s − 1.18·13-s + 2.01·14-s + 16-s − 2.84·17-s + 6.39·19-s − 2.40·20-s − 4.74·22-s + 3.82·23-s + 0.785·25-s − 1.18·26-s + 2.01·28-s − 1.12·29-s + 4.42·31-s + 32-s − 2.84·34-s − 4.84·35-s − 4.49·37-s + 6.39·38-s − 2.40·40-s − 12.3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.07·5-s + 0.761·7-s + 0.353·8-s − 0.760·10-s − 1.42·11-s − 0.329·13-s + 0.538·14-s + 0.250·16-s − 0.690·17-s + 1.46·19-s − 0.537·20-s − 1.01·22-s + 0.796·23-s + 0.157·25-s − 0.233·26-s + 0.380·28-s − 0.209·29-s + 0.795·31-s + 0.176·32-s − 0.488·34-s − 0.818·35-s − 0.739·37-s + 1.03·38-s − 0.380·40-s − 1.93·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 - 2.01T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 19 | \( 1 - 6.39T + 19T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 + 4.49T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 + 2.24T + 43T^{2} \) |
| 47 | \( 1 - 3.85T + 47T^{2} \) |
| 53 | \( 1 + 8.02T + 53T^{2} \) |
| 59 | \( 1 + 8.06T + 59T^{2} \) |
| 61 | \( 1 + 0.752T + 61T^{2} \) |
| 67 | \( 1 + 5.18T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 1.50T + 79T^{2} \) |
| 83 | \( 1 + 2.56T + 83T^{2} \) |
| 89 | \( 1 + 8.08T + 89T^{2} \) |
| 97 | \( 1 - 2.17T + 97T^{2} \) |
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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
−7.957605404160542155683667132838, −7.39224525696560411736523577786, −6.75060035300481606184524510188, −5.53049102736003147178346637698, −4.98509938568050514557777223497, −4.45051039165186751977238154506, −3.37049355893131588760479551436, −2.77411492682191424164672791509, −1.55695054665038699785426099275, 0,
1.55695054665038699785426099275, 2.77411492682191424164672791509, 3.37049355893131588760479551436, 4.45051039165186751977238154506, 4.98509938568050514557777223497, 5.53049102736003147178346637698, 6.75060035300481606184524510188, 7.39224525696560411736523577786, 7.957605404160542155683667132838
