| L(s) = 1 | + 2.51·3-s + 0.302·5-s − 3.27·7-s + 3.30·9-s − 0.760·11-s + 5.90·13-s + 0.760·15-s − 2.51·19-s − 8.21·21-s − 3.27·23-s − 4.90·25-s + 0.760·27-s − 7.30·29-s + 0.760·31-s − 1.90·33-s − 0.990·35-s − 4·37-s + 14.8·39-s − 5.21·41-s + 10.8·43-s + 1.00·45-s − 9.28·47-s + 3.69·49-s − 2.09·53-s − 0.230·55-s − 6.30·57-s + 13.0·59-s + ⋯ |
| L(s) = 1 | + 1.44·3-s + 0.135·5-s − 1.23·7-s + 1.10·9-s − 0.229·11-s + 1.63·13-s + 0.196·15-s − 0.575·19-s − 1.79·21-s − 0.681·23-s − 0.981·25-s + 0.146·27-s − 1.35·29-s + 0.136·31-s − 0.332·33-s − 0.167·35-s − 0.657·37-s + 2.37·39-s − 0.813·41-s + 1.64·43-s + 0.149·45-s − 1.35·47-s + 0.528·49-s − 0.287·53-s − 0.0310·55-s − 0.834·57-s + 1.70·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2.51T + 3T^{2} \) |
| 5 | \( 1 - 0.302T + 5T^{2} \) |
| 7 | \( 1 + 3.27T + 7T^{2} \) |
| 11 | \( 1 + 0.760T + 11T^{2} \) |
| 13 | \( 1 - 5.90T + 13T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 + 3.27T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 - 0.760T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 5.21T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 9.28T + 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 9.21T + 89T^{2} \) |
| 97 | \( 1 - 1.51T + 97T^{2} \) |
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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
−7.55136281044864236796945839313, −6.68919664159585102899418555070, −6.12299807229440494170976985402, −5.47576507150185901664982955170, −4.10497643162246986284061540478, −3.71692763219783463287074893044, −3.14722930603667081202463390783, −2.28860168945865854548131264823, −1.53748178592283272500342161625, 0,
1.53748178592283272500342161625, 2.28860168945865854548131264823, 3.14722930603667081202463390783, 3.71692763219783463287074893044, 4.10497643162246986284061540478, 5.47576507150185901664982955170, 6.12299807229440494170976985402, 6.68919664159585102899418555070, 7.55136281044864236796945839313
