| L(s) = 1 | − 2.38·3-s − 1.71·5-s + 3.80·7-s + 2.68·9-s + 11-s + 0.388·13-s + 4.09·15-s + 4.85·17-s − 2.41·19-s − 9.07·21-s + 2.68·23-s − 2.04·25-s + 0.747·27-s − 0.379·29-s + 5.96·31-s − 2.38·33-s − 6.54·35-s + 10.4·37-s − 0.925·39-s + 5.04·41-s + 0.468·43-s − 4.61·45-s − 5.27·47-s + 7.48·49-s − 11.5·51-s + 2.39·53-s − 1.71·55-s + ⋯ |
| L(s) = 1 | − 1.37·3-s − 0.768·5-s + 1.43·7-s + 0.895·9-s + 0.301·11-s + 0.107·13-s + 1.05·15-s + 1.17·17-s − 0.554·19-s − 1.98·21-s + 0.559·23-s − 0.409·25-s + 0.143·27-s − 0.0705·29-s + 1.07·31-s − 0.415·33-s − 1.10·35-s + 1.71·37-s − 0.148·39-s + 0.787·41-s + 0.0714·43-s − 0.688·45-s − 0.770·47-s + 1.06·49-s − 1.62·51-s + 0.328·53-s − 0.231·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.279246455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.279246455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 + 2.38T + 3T^{2} \) |
| 5 | \( 1 + 1.71T + 5T^{2} \) |
| 7 | \( 1 - 3.80T + 7T^{2} \) |
| 13 | \( 1 - 0.388T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 + 0.379T + 29T^{2} \) |
| 31 | \( 1 - 5.96T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 5.04T + 41T^{2} \) |
| 43 | \( 1 - 0.468T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 - 2.39T + 53T^{2} \) |
| 59 | \( 1 - 1.80T + 59T^{2} \) |
| 61 | \( 1 - 5.26T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 6.64T + 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 + 0.272T + 83T^{2} \) |
| 89 | \( 1 + 2.59T + 89T^{2} \) |
| 97 | \( 1 + 4.65T + 97T^{2} \) |
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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
−7.994714953461992370563817213596, −7.42987570542601702876508341380, −6.56380841479582862434217315586, −5.84257949765054552272770014237, −5.23048940892599719389120412970, −4.50881259053221707935996614065, −4.02621242445628083696071933351, −2.77572702112363409303285288948, −1.46200392064710154796009912485, −0.70631612409190956423132203840,
0.70631612409190956423132203840, 1.46200392064710154796009912485, 2.77572702112363409303285288948, 4.02621242445628083696071933351, 4.50881259053221707935996614065, 5.23048940892599719389120412970, 5.84257949765054552272770014237, 6.56380841479582862434217315586, 7.42987570542601702876508341380, 7.994714953461992370563817213596
