L(s) = 1 | + 2.40·3-s + 0.590·5-s − 1.95·7-s + 2.78·9-s + 1.52·11-s − 4.67·13-s + 1.41·15-s − 3.04·17-s − 0.338·19-s − 4.71·21-s + 7.98·23-s − 4.65·25-s − 0.513·27-s + 6.01·29-s + 4.17·31-s + 3.67·33-s − 1.15·35-s − 11.0·37-s − 11.2·39-s − 4.82·41-s − 12.4·43-s + 1.64·45-s − 11.0·47-s − 3.16·49-s − 7.33·51-s − 3.43·53-s + 0.901·55-s + ⋯ |
L(s) = 1 | + 1.38·3-s + 0.263·5-s − 0.740·7-s + 0.928·9-s + 0.460·11-s − 1.29·13-s + 0.366·15-s − 0.739·17-s − 0.0775·19-s − 1.02·21-s + 1.66·23-s − 0.930·25-s − 0.0988·27-s + 1.11·29-s + 0.749·31-s + 0.639·33-s − 0.195·35-s − 1.81·37-s − 1.80·39-s − 0.753·41-s − 1.89·43-s + 0.245·45-s − 1.60·47-s − 0.451·49-s − 1.02·51-s − 0.472·53-s + 0.121·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 - 0.590T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 19 | \( 1 + 0.338T + 19T^{2} \) |
| 23 | \( 1 - 7.98T + 23T^{2} \) |
| 29 | \( 1 - 6.01T + 29T^{2} \) |
| 31 | \( 1 - 4.17T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 3.43T + 53T^{2} \) |
| 59 | \( 1 + 2.63T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 9.66T + 71T^{2} \) |
| 73 | \( 1 + 6.33T + 73T^{2} \) |
| 79 | \( 1 - 0.667T + 79T^{2} \) |
| 83 | \( 1 - 2.26T + 83T^{2} \) |
| 89 | \( 1 - 2.21T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.49722904375810744840002044630, −6.77542056464147596411211759338, −6.48266444309668303799997857326, −5.15651086161379366626457229209, −4.68240091146949948276119229558, −3.55822240707759241407342831285, −3.12247533371052104727667212527, −2.36857156174397122903587858360, −1.58510729611366036278814425202, 0,
1.58510729611366036278814425202, 2.36857156174397122903587858360, 3.12247533371052104727667212527, 3.55822240707759241407342831285, 4.68240091146949948276119229558, 5.15651086161379366626457229209, 6.48266444309668303799997857326, 6.77542056464147596411211759338, 7.49722904375810744840002044630