| L(s) = 1 | + 5.90e4·3-s + 4.18e7·5-s + 6.87e8·7-s + 3.48e9·9-s − 1.49e9·11-s − 8.84e9·13-s + 2.46e12·15-s − 9.41e12·17-s + 4.23e13·19-s + 4.05e13·21-s − 1.09e14·23-s + 1.27e15·25-s + 2.05e14·27-s + 1.18e15·29-s + 1.95e15·31-s − 8.84e13·33-s + 2.87e16·35-s + 4.20e16·37-s − 5.22e14·39-s − 6.92e16·41-s − 8.95e16·43-s + 1.45e17·45-s + 5.26e17·47-s − 8.63e16·49-s − 5.55e17·51-s − 2.24e18·53-s − 6.26e16·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.91·5-s + 0.919·7-s + 0.333·9-s − 0.0174·11-s − 0.0178·13-s + 1.10·15-s − 1.13·17-s + 1.58·19-s + 0.530·21-s − 0.550·23-s + 2.66·25-s + 0.192·27-s + 0.521·29-s + 0.427·31-s − 0.0100·33-s + 1.76·35-s + 1.43·37-s − 0.0102·39-s − 0.805·41-s − 0.631·43-s + 0.638·45-s + 1.46·47-s − 0.154·49-s − 0.653·51-s − 1.76·53-s − 0.0333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(5.288269453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.288269453\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.90e4T \) |
| good | 5 | \( 1 - 4.18e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 6.87e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.49e9T + 7.40e21T^{2} \) |
| 13 | \( 1 + 8.84e9T + 2.47e23T^{2} \) |
| 17 | \( 1 + 9.41e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 4.23e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.09e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.18e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 1.95e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.20e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 6.92e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 8.95e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 5.26e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 2.24e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.41e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 3.17e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.04e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.92e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 4.75e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 8.63e18T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.65e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.57e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 9.21e20T + 5.27e41T^{2} \) |
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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
−11.34300238123302376929463052021, −10.08017398328071519605753876719, −9.330332239168489899053286200731, −8.225718325158488944157989291687, −6.79141561668127672117376114568, −5.62090036582903916846820036347, −4.62127385176339780259339210210, −2.83337314166100481793807094223, −1.95029995184043274953225468250, −1.11775244235263744861509416709,
1.11775244235263744861509416709, 1.95029995184043274953225468250, 2.83337314166100481793807094223, 4.62127385176339780259339210210, 5.62090036582903916846820036347, 6.79141561668127672117376114568, 8.225718325158488944157989291687, 9.330332239168489899053286200731, 10.08017398328071519605753876719, 11.34300238123302376929463052021
