| L(s) = 1 | − 1.95·3-s + 3.19·5-s + 4.95·7-s + 0.815·9-s − 0.512·11-s + 6.09·13-s − 6.24·15-s − 5.31·17-s + 5.55·19-s − 9.67·21-s − 0.279·23-s + 5.23·25-s + 4.26·27-s + 0.455·29-s − 3.23·31-s + 1.00·33-s + 15.8·35-s − 8.61·37-s − 11.9·39-s + 1.45·41-s + 5.88·43-s + 2.60·45-s + 6.09·47-s + 17.5·49-s + 10.3·51-s + 1.06·53-s − 1.64·55-s + ⋯ |
| L(s) = 1 | − 1.12·3-s + 1.43·5-s + 1.87·7-s + 0.271·9-s − 0.154·11-s + 1.68·13-s − 1.61·15-s − 1.28·17-s + 1.27·19-s − 2.11·21-s − 0.0582·23-s + 1.04·25-s + 0.821·27-s + 0.0845·29-s − 0.581·31-s + 0.174·33-s + 2.67·35-s − 1.41·37-s − 1.90·39-s + 0.227·41-s + 0.898·43-s + 0.388·45-s + 0.888·47-s + 2.50·49-s + 1.45·51-s + 0.145·53-s − 0.221·55-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
| \(L(1)\) |
\(\approx\) |
\(2.529402498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.529402498\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 7 | \( 1 - 4.95T + 7T^{2} \) |
| 11 | \( 1 + 0.512T + 11T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 + 5.31T + 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 23 | \( 1 + 0.279T + 23T^{2} \) |
| 29 | \( 1 - 0.455T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 - 5.88T + 43T^{2} \) |
| 47 | \( 1 - 6.09T + 47T^{2} \) |
| 53 | \( 1 - 1.06T + 53T^{2} \) |
| 59 | \( 1 - 4.58T + 59T^{2} \) |
| 61 | \( 1 - 0.389T + 61T^{2} \) |
| 67 | \( 1 + 0.0619T + 67T^{2} \) |
| 71 | \( 1 - 3.23T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 3.20T + 79T^{2} \) |
| 83 | \( 1 - 1.53T + 83T^{2} \) |
| 89 | \( 1 - 3.21T + 89T^{2} \) |
| 97 | \( 1 + 9.63T + 97T^{2} \) |
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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
−8.202483684987365986102305333982, −7.19475142462444266704822437239, −6.48428651197784638554304342802, −5.66999821979031707312431852540, −5.45512792245971090381351397261, −4.77378877321581762356521287523, −3.85958495737219210595279917954, −2.46417028842423746393705880129, −1.64032518157185299692773225679, −0.987265974707598993109319626078,
0.987265974707598993109319626078, 1.64032518157185299692773225679, 2.46417028842423746393705880129, 3.85958495737219210595279917954, 4.77378877321581762356521287523, 5.45512792245971090381351397261, 5.66999821979031707312431852540, 6.48428651197784638554304342802, 7.19475142462444266704822437239, 8.202483684987365986102305333982
