| L(s) = 1 | + 5.99e5·3-s − 1.94e8·5-s + 4.67e9·7-s + 2.65e11·9-s + 9.07e11·11-s + 6.04e12·13-s − 1.16e14·15-s − 2.80e13·17-s − 9.06e13·19-s + 2.80e15·21-s + 2.89e14·23-s + 2.57e16·25-s + 1.02e17·27-s + 4.09e16·29-s − 1.20e17·31-s + 5.44e17·33-s − 9.07e17·35-s − 1.40e18·37-s + 3.62e18·39-s + 5.50e17·41-s + 8.04e18·43-s − 5.15e19·45-s + 2.67e18·47-s − 5.50e18·49-s − 1.68e19·51-s − 4.15e19·53-s − 1.76e20·55-s + ⋯ |
| L(s) = 1 | + 1.95·3-s − 1.77·5-s + 0.893·7-s + 2.82·9-s + 0.959·11-s + 0.935·13-s − 3.47·15-s − 0.198·17-s − 0.178·19-s + 1.74·21-s + 0.0633·23-s + 2.16·25-s + 3.56·27-s + 0.622·29-s − 0.853·31-s + 1.87·33-s − 1.58·35-s − 1.29·37-s + 1.82·39-s + 0.156·41-s + 1.32·43-s − 5.01·45-s + 0.157·47-s − 0.201·49-s − 0.387·51-s − 0.615·53-s − 1.70·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(5.143723314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.143723314\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 5.99e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.94e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 4.67e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 9.07e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 6.04e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.80e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 9.06e13T + 2.57e29T^{2} \) |
| 23 | \( 1 - 2.89e14T + 2.08e31T^{2} \) |
| 29 | \( 1 - 4.09e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.20e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.40e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 5.50e17T + 1.24e37T^{2} \) |
| 43 | \( 1 - 8.04e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 2.67e18T + 2.87e38T^{2} \) |
| 53 | \( 1 + 4.15e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.95e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 3.54e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.08e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.49e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.40e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 9.78e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 2.67e20T + 1.37e44T^{2} \) |
| 89 | \( 1 - 1.67e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 9.64e22T + 4.96e45T^{2} \) |
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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
−10.67505116698284593184931161848, −9.027133671701047411806795010852, −8.494885650420800231135096375130, −7.74099738547631749968574827795, −6.92316914130500567261691824174, −4.51567759524279955905697519220, −3.88391425076264117135871577842, −3.19113633488599188702313388408, −1.84036911239910325742175419463, −0.907404204769306454730164192475,
0.907404204769306454730164192475, 1.84036911239910325742175419463, 3.19113633488599188702313388408, 3.88391425076264117135871577842, 4.51567759524279955905697519220, 6.92316914130500567261691824174, 7.74099738547631749968574827795, 8.494885650420800231135096375130, 9.027133671701047411806795010852, 10.67505116698284593184931161848
