| L(s) = 1 | − 0.0692·3-s + 0.775·5-s − 4.19·7-s − 2.99·9-s − 0.210·11-s − 5.86·13-s − 0.0536·15-s − 6.81·17-s − 7.37·19-s + 0.290·21-s + 3.64·23-s − 4.39·25-s + 0.414·27-s − 6.37·29-s − 8.26·31-s + 0.0145·33-s − 3.25·35-s + 5.38·37-s + 0.405·39-s + 7.37·41-s − 4.77·43-s − 2.32·45-s − 7.67·47-s + 10.6·49-s + 0.471·51-s − 4.45·53-s − 0.163·55-s + ⋯ |
| L(s) = 1 | − 0.0399·3-s + 0.346·5-s − 1.58·7-s − 0.998·9-s − 0.0634·11-s − 1.62·13-s − 0.0138·15-s − 1.65·17-s − 1.69·19-s + 0.0634·21-s + 0.759·23-s − 0.879·25-s + 0.0798·27-s − 1.18·29-s − 1.48·31-s + 0.00253·33-s − 0.550·35-s + 0.884·37-s + 0.0650·39-s + 1.15·41-s − 0.728·43-s − 0.346·45-s − 1.11·47-s + 1.51·49-s + 0.0660·51-s − 0.611·53-s − 0.0220·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04663874971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04663874971\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0692T + 3T^{2} \) |
| 5 | \( 1 - 0.775T + 5T^{2} \) |
| 7 | \( 1 + 4.19T + 7T^{2} \) |
| 11 | \( 1 + 0.210T + 11T^{2} \) |
| 13 | \( 1 + 5.86T + 13T^{2} \) |
| 17 | \( 1 + 6.81T + 17T^{2} \) |
| 19 | \( 1 + 7.37T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 + 6.37T + 29T^{2} \) |
| 31 | \( 1 + 8.26T + 31T^{2} \) |
| 37 | \( 1 - 5.38T + 37T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 + 4.77T + 43T^{2} \) |
| 47 | \( 1 + 7.67T + 47T^{2} \) |
| 53 | \( 1 + 4.45T + 53T^{2} \) |
| 59 | \( 1 - 3.83T + 59T^{2} \) |
| 61 | \( 1 - 9.08T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 - 8.49T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 3.41T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 9.76T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.79337095612505325468313693841, −6.87948428453507248873510320649, −6.56267860407555093659115201797, −5.82043650244109185431890433068, −5.15155446399262536396710591035, −4.22568628642220447604432043962, −3.47491426997062385134513523861, −2.41590181642225089024067504909, −2.24180458188326647204476801645, −0.095519350768958260534629813596,
0.095519350768958260534629813596, 2.24180458188326647204476801645, 2.41590181642225089024067504909, 3.47491426997062385134513523861, 4.22568628642220447604432043962, 5.15155446399262536396710591035, 5.82043650244109185431890433068, 6.56267860407555093659115201797, 6.87948428453507248873510320649, 7.79337095612505325468313693841
