| L(s) = 1 | − 2.87·3-s − 0.461·5-s − 2.48·7-s + 5.25·9-s − 0.455·11-s − 4.37·13-s + 1.32·15-s + 17-s + 4.83·19-s + 7.13·21-s + 4.61·23-s − 4.78·25-s − 6.49·27-s + 1.04·29-s + 6.92·31-s + 1.30·33-s + 1.14·35-s − 7.50·37-s + 12.5·39-s − 7.24·41-s + 0.668·43-s − 2.42·45-s − 10.7·47-s − 0.839·49-s − 2.87·51-s − 1.23·53-s + 0.210·55-s + ⋯ |
| L(s) = 1 | − 1.65·3-s − 0.206·5-s − 0.938·7-s + 1.75·9-s − 0.137·11-s − 1.21·13-s + 0.342·15-s + 0.242·17-s + 1.10·19-s + 1.55·21-s + 0.961·23-s − 0.957·25-s − 1.24·27-s + 0.194·29-s + 1.24·31-s + 0.227·33-s + 0.193·35-s − 1.23·37-s + 2.01·39-s − 1.13·41-s + 0.101·43-s − 0.361·45-s − 1.56·47-s − 0.119·49-s − 0.402·51-s − 0.169·53-s + 0.0283·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4188346665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4188346665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.87T + 3T^{2} \) |
| 5 | \( 1 + 0.461T + 5T^{2} \) |
| 7 | \( 1 + 2.48T + 7T^{2} \) |
| 11 | \( 1 + 0.455T + 11T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 - 1.04T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 7.50T + 37T^{2} \) |
| 41 | \( 1 + 7.24T + 41T^{2} \) |
| 43 | \( 1 - 0.668T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 3.21T + 67T^{2} \) |
| 71 | \( 1 - 2.39T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 1.92T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−7.55732078664117648451759708797, −6.88027999930921498223010007311, −6.56593451823555483520246758220, −5.63722081404766998455810937553, −5.16152371204854334451194402517, −4.59357862625944358119646189601, −3.54705737939846258425374487033, −2.78225694854959378741086281607, −1.44392879467341833838353997592, −0.36401535662765848668420125476,
0.36401535662765848668420125476, 1.44392879467341833838353997592, 2.78225694854959378741086281607, 3.54705737939846258425374487033, 4.59357862625944358119646189601, 5.16152371204854334451194402517, 5.63722081404766998455810937553, 6.56593451823555483520246758220, 6.88027999930921498223010007311, 7.55732078664117648451759708797
