| L(s) = 1 | + 0.621·3-s − 0.558·5-s − 3.11·7-s − 2.61·9-s − 4.48·11-s − 4.38·13-s − 0.347·15-s + 0.883·17-s + 0.759·19-s − 1.93·21-s − 8.56·23-s − 4.68·25-s − 3.48·27-s + 3.34·29-s + 8.74·31-s − 2.78·33-s + 1.74·35-s − 1.36·37-s − 2.72·39-s − 4.23·41-s + 6.52·43-s + 1.45·45-s − 4.29·47-s + 2.71·49-s + 0.549·51-s + 5.36·53-s + 2.50·55-s + ⋯ |
| L(s) = 1 | + 0.358·3-s − 0.249·5-s − 1.17·7-s − 0.871·9-s − 1.35·11-s − 1.21·13-s − 0.0896·15-s + 0.214·17-s + 0.174·19-s − 0.422·21-s − 1.78·23-s − 0.937·25-s − 0.671·27-s + 0.620·29-s + 1.57·31-s − 0.485·33-s + 0.294·35-s − 0.223·37-s − 0.435·39-s − 0.661·41-s + 0.995·43-s + 0.217·45-s − 0.625·47-s + 0.388·49-s + 0.0768·51-s + 0.737·53-s + 0.337·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5306660157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5306660157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 0.621T + 3T^{2} \) |
| 5 | \( 1 + 0.558T + 5T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 13 | \( 1 + 4.38T + 13T^{2} \) |
| 17 | \( 1 - 0.883T + 17T^{2} \) |
| 19 | \( 1 - 0.759T + 19T^{2} \) |
| 23 | \( 1 + 8.56T + 23T^{2} \) |
| 29 | \( 1 - 3.34T + 29T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 - 6.52T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 - 5.36T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 - 7.13T + 67T^{2} \) |
| 71 | \( 1 + 4.99T + 71T^{2} \) |
| 73 | \( 1 + 3.75T + 73T^{2} \) |
| 79 | \( 1 + 2.44T + 79T^{2} \) |
| 83 | \( 1 + 2.13T + 83T^{2} \) |
| 89 | \( 1 - 2.36T + 89T^{2} \) |
| 97 | \( 1 - 1.03T + 97T^{2} \) |
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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
−7.993470465490624739994429492183, −7.59183082317931160411763797461, −6.64429544985733619473778954593, −5.92259983544972027001015163475, −5.31494639686374700624898720674, −4.38934424260224226161638750196, −3.46983807222938997224033115299, −2.73742660592101661066162466994, −2.24329014603617586501749308726, −0.34198782146044536377813833925,
0.34198782146044536377813833925, 2.24329014603617586501749308726, 2.73742660592101661066162466994, 3.46983807222938997224033115299, 4.38934424260224226161638750196, 5.31494639686374700624898720674, 5.92259983544972027001015163475, 6.64429544985733619473778954593, 7.59183082317931160411763797461, 7.993470465490624739994429492183
