| L(s) = 1 | + 1.76·3-s + 5-s + 0.0387·7-s + 0.122·9-s − 4.56·11-s + 2.95·13-s + 1.76·15-s − 3.20·17-s + 2.27·19-s + 0.0685·21-s + 5.32·23-s + 25-s − 5.08·27-s − 9.46·29-s − 8.25·31-s − 8.07·33-s + 0.0387·35-s − 8.02·37-s + 5.22·39-s − 1.45·41-s − 4.90·43-s + 0.122·45-s − 6.28·47-s − 6.99·49-s − 5.65·51-s − 0.800·53-s − 4.56·55-s + ⋯ |
| L(s) = 1 | + 1.02·3-s + 0.447·5-s + 0.0146·7-s + 0.0408·9-s − 1.37·11-s + 0.820·13-s + 0.456·15-s − 0.776·17-s + 0.522·19-s + 0.0149·21-s + 1.11·23-s + 0.200·25-s − 0.978·27-s − 1.75·29-s − 1.48·31-s − 1.40·33-s + 0.00655·35-s − 1.31·37-s + 0.837·39-s − 0.227·41-s − 0.747·43-s + 0.0182·45-s − 0.916·47-s − 0.999·49-s − 0.792·51-s − 0.110·53-s − 0.616·55-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 | \( 1 - 1.76T + 3T^{2} \) |
| 7 | \( 1 - 0.0387T + 7T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 13 | \( 1 - 2.95T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 - 5.32T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 + 8.25T + 31T^{2} \) |
| 37 | \( 1 + 8.02T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 + 4.90T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 + 0.800T + 53T^{2} \) |
| 59 | \( 1 + 7.10T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 2.60T + 83T^{2} \) |
| 89 | \( 1 + 1.97T + 89T^{2} \) |
| 97 | \( 1 - 6.49T + 97T^{2} \) |
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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
−7.83811297241010953493624726345, −7.17551668568137699970516772941, −6.36128946714073868070235652953, −5.34757963044655640905909114863, −5.08447249167320734230836130072, −3.66732300025183849265516371532, −3.27977980662433520073109608007, −2.31668929075883667215564078898, −1.67294659163830924722304616437, 0,
1.67294659163830924722304616437, 2.31668929075883667215564078898, 3.27977980662433520073109608007, 3.66732300025183849265516371532, 5.08447249167320734230836130072, 5.34757963044655640905909114863, 6.36128946714073868070235652953, 7.17551668568137699970516772941, 7.83811297241010953493624726345
