| L(s) = 1 | + 0.565·3-s − 5-s + 3.63·7-s − 2.68·9-s − 2.11·11-s − 0.576·13-s − 0.565·15-s − 0.572·17-s + 1.12·19-s + 2.05·21-s − 0.0407·23-s + 25-s − 3.21·27-s + 3.94·29-s − 6.96·31-s − 1.19·33-s − 3.63·35-s + 4.29·37-s − 0.326·39-s − 9.86·41-s + 8.38·43-s + 2.68·45-s − 3.46·47-s + 6.24·49-s − 0.323·51-s − 6.50·53-s + 2.11·55-s + ⋯ |
| L(s) = 1 | + 0.326·3-s − 0.447·5-s + 1.37·7-s − 0.893·9-s − 0.638·11-s − 0.159·13-s − 0.145·15-s − 0.138·17-s + 0.258·19-s + 0.448·21-s − 0.00850·23-s + 0.200·25-s − 0.618·27-s + 0.732·29-s − 1.25·31-s − 0.208·33-s − 0.615·35-s + 0.706·37-s − 0.0522·39-s − 1.54·41-s + 1.27·43-s + 0.399·45-s − 0.504·47-s + 0.891·49-s − 0.0453·51-s − 0.893·53-s + 0.285·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 0.565T + 3T^{2} \) |
| 7 | \( 1 - 3.63T + 7T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 13 | \( 1 + 0.576T + 13T^{2} \) |
| 17 | \( 1 + 0.572T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 0.0407T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 + 6.96T + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 + 9.86T + 41T^{2} \) |
| 43 | \( 1 - 8.38T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 6.50T + 53T^{2} \) |
| 59 | \( 1 + 3.93T + 59T^{2} \) |
| 61 | \( 1 + 3.94T + 61T^{2} \) |
| 67 | \( 1 + 0.484T + 67T^{2} \) |
| 71 | \( 1 + 5.45T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 2.17T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.84148730867761683001520080575, −7.29047424673116457221002490655, −6.26621852421925175876632108756, −5.37750202643047925542962956046, −4.91830780542294583893328582691, −4.08114077794526505549402927687, −3.13627385780560845066457275558, −2.38209812245136938681045839669, −1.41400093420341802343025764761, 0,
1.41400093420341802343025764761, 2.38209812245136938681045839669, 3.13627385780560845066457275558, 4.08114077794526505549402927687, 4.91830780542294583893328582691, 5.37750202643047925542962956046, 6.26621852421925175876632108756, 7.29047424673116457221002490655, 7.84148730867761683001520080575
