| L(s) = 1 | + 1.89·3-s − 2.72·5-s + 0.607·9-s + 1.72·11-s + 13-s − 5.17·15-s + 7.29·17-s − 7.05·19-s − 6.84·23-s + 2.41·25-s − 4.54·27-s − 1.28·29-s − 0.275·31-s + 3.27·33-s + 8.31·37-s + 1.89·39-s + 4.18·41-s + 1.82·43-s − 1.65·45-s − 8.10·47-s + 13.8·51-s − 9.66·53-s − 4.68·55-s − 13.3·57-s − 4.82·59-s − 12.1·61-s − 2.72·65-s + ⋯ |
| L(s) = 1 | + 1.09·3-s − 1.21·5-s + 0.202·9-s + 0.519·11-s + 0.277·13-s − 1.33·15-s + 1.77·17-s − 1.61·19-s − 1.42·23-s + 0.482·25-s − 0.874·27-s − 0.239·29-s − 0.0494·31-s + 0.569·33-s + 1.36·37-s + 0.304·39-s + 0.653·41-s + 0.277·43-s − 0.246·45-s − 1.18·47-s + 1.94·51-s − 1.32·53-s − 0.632·55-s − 1.77·57-s − 0.627·59-s − 1.56·61-s − 0.337·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.89T + 3T^{2} \) |
| 5 | \( 1 + 2.72T + 5T^{2} \) |
| 11 | \( 1 - 1.72T + 11T^{2} \) |
| 17 | \( 1 - 7.29T + 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 + 1.28T + 29T^{2} \) |
| 31 | \( 1 + 0.275T + 31T^{2} \) |
| 37 | \( 1 - 8.31T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 + 9.66T + 53T^{2} \) |
| 59 | \( 1 + 4.82T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 0.111T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 4.75T + 79T^{2} \) |
| 83 | \( 1 - 8.34T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 2.79T + 97T^{2} \) |
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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
−7.84442229637859917461013671456, −7.66498826974309258492785347436, −6.42128190767943706494314678062, −5.85648169181654694444274313623, −4.58862714072647000984146364829, −3.89891798235727639659151618804, −3.45287252720643283084369868450, −2.54706150947751997533227130610, −1.47797751030311333808528288704, 0,
1.47797751030311333808528288704, 2.54706150947751997533227130610, 3.45287252720643283084369868450, 3.89891798235727639659151618804, 4.58862714072647000984146364829, 5.85648169181654694444274313623, 6.42128190767943706494314678062, 7.66498826974309258492785347436, 7.84442229637859917461013671456
