| L(s) = 1 | + 0.438·3-s + 3.19·5-s − 3.98·7-s − 2.80·9-s − 11-s + 5.23·13-s + 1.39·15-s + 2.24·17-s + 19-s − 1.74·21-s + 2.71·23-s + 5.18·25-s − 2.54·27-s + 6.72·29-s + 4.03·31-s − 0.438·33-s − 12.7·35-s − 5.79·37-s + 2.29·39-s − 6.31·41-s − 0.982·43-s − 8.96·45-s + 2.38·47-s + 8.90·49-s + 0.985·51-s + 0.773·53-s − 3.19·55-s + ⋯ |
| L(s) = 1 | + 0.252·3-s + 1.42·5-s − 1.50·7-s − 0.936·9-s − 0.301·11-s + 1.45·13-s + 0.360·15-s + 0.545·17-s + 0.229·19-s − 0.381·21-s + 0.565·23-s + 1.03·25-s − 0.489·27-s + 1.24·29-s + 0.725·31-s − 0.0762·33-s − 2.15·35-s − 0.952·37-s + 0.367·39-s − 0.986·41-s − 0.149·43-s − 1.33·45-s + 0.347·47-s + 1.27·49-s + 0.138·51-s + 0.106·53-s − 0.430·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.236102748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.236102748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.438T + 3T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 29 | \( 1 - 6.72T + 29T^{2} \) |
| 31 | \( 1 - 4.03T + 31T^{2} \) |
| 37 | \( 1 + 5.79T + 37T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 + 0.982T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 - 0.773T + 53T^{2} \) |
| 59 | \( 1 + 2.37T + 59T^{2} \) |
| 61 | \( 1 - 1.97T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 8.69T + 71T^{2} \) |
| 73 | \( 1 + 0.457T + 73T^{2} \) |
| 79 | \( 1 - 1.37T + 79T^{2} \) |
| 83 | \( 1 + 2.14T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.781660455162132563622309379643, −8.098245088972798747497376930832, −6.78662369078257082208851241664, −6.34780548447316915838243186637, −5.75622076169346959752502285020, −5.06475536595620170954237698722, −3.54428799648318420093673711130, −3.08776385338385622200408068724, −2.18653798125076845500030349915, −0.883895073871911066502301519748,
0.883895073871911066502301519748, 2.18653798125076845500030349915, 3.08776385338385622200408068724, 3.54428799648318420093673711130, 5.06475536595620170954237698722, 5.75622076169346959752502285020, 6.34780548447316915838243186637, 6.78662369078257082208851241664, 8.098245088972798747497376930832, 8.781660455162132563622309379643
