| L(s) = 1 | + 4.98e6·3-s + 3.10e9·5-s + 2.36e11·7-s + 1.72e13·9-s + 2.12e14·11-s − 1.46e15·13-s + 1.54e16·15-s + 2.34e15·17-s + 9.03e16·19-s + 1.17e18·21-s − 1.52e18·23-s + 2.18e18·25-s + 4.77e19·27-s + 1.80e19·29-s − 3.08e19·31-s + 1.05e21·33-s + 7.33e20·35-s − 6.43e20·37-s − 7.32e21·39-s − 7.21e21·41-s − 1.26e22·43-s + 5.34e22·45-s − 6.23e22·47-s − 9.79e21·49-s + 1.16e22·51-s + 1.22e23·53-s + 6.59e23·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s + 1.13·5-s + 0.922·7-s + 2.25·9-s + 1.85·11-s − 1.34·13-s + 2.05·15-s + 0.0574·17-s + 0.492·19-s + 1.66·21-s − 0.631·23-s + 0.292·25-s + 2.26·27-s + 0.326·29-s − 0.226·31-s + 3.34·33-s + 1.04·35-s − 0.434·37-s − 2.42·39-s − 1.21·41-s − 1.11·43-s + 2.56·45-s − 1.66·47-s − 0.149·49-s + 0.103·51-s + 0.646·53-s + 2.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(\approx\) |
\(6.469567215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.469567215\) |
| \(L(\frac{29}{2})\) |
|
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 | \( 1 - 4.98e6T + 7.62e12T^{2} \) |
| 5 | \( 1 - 3.10e9T + 7.45e18T^{2} \) |
| 7 | \( 1 - 2.36e11T + 6.57e22T^{2} \) |
| 11 | \( 1 - 2.12e14T + 1.31e28T^{2} \) |
| 13 | \( 1 + 1.46e15T + 1.19e30T^{2} \) |
| 17 | \( 1 - 2.34e15T + 1.66e33T^{2} \) |
| 19 | \( 1 - 9.03e16T + 3.36e34T^{2} \) |
| 23 | \( 1 + 1.52e18T + 5.84e36T^{2} \) |
| 29 | \( 1 - 1.80e19T + 3.05e39T^{2} \) |
| 31 | \( 1 + 3.08e19T + 1.84e40T^{2} \) |
| 37 | \( 1 + 6.43e20T + 2.19e42T^{2} \) |
| 41 | \( 1 + 7.21e21T + 3.50e43T^{2} \) |
| 43 | \( 1 + 1.26e22T + 1.26e44T^{2} \) |
| 47 | \( 1 + 6.23e22T + 1.40e45T^{2} \) |
| 53 | \( 1 - 1.22e23T + 3.59e46T^{2} \) |
| 59 | \( 1 + 1.26e24T + 6.50e47T^{2} \) |
| 61 | \( 1 + 4.90e23T + 1.59e48T^{2} \) |
| 67 | \( 1 - 2.60e24T + 2.01e49T^{2} \) |
| 71 | \( 1 + 7.66e24T + 9.63e49T^{2} \) |
| 73 | \( 1 - 2.19e25T + 2.04e50T^{2} \) |
| 79 | \( 1 - 7.76e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + 3.00e25T + 6.53e51T^{2} \) |
| 89 | \( 1 - 2.29e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 1.71e26T + 4.39e53T^{2} \) |
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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
−13.65135218437225710582321850655, −12.03413052079825843817815850242, −9.907111154658323456103239024973, −9.243636028279992175944653782056, −8.067270190111903599253911599863, −6.74721177642138430341489504585, −4.79051127184189824090222370309, −3.42314905000503201273184687042, −2.03103965597371932695249098477, −1.51508517941992458298148864352,
1.51508517941992458298148864352, 2.03103965597371932695249098477, 3.42314905000503201273184687042, 4.79051127184189824090222370309, 6.74721177642138430341489504585, 8.067270190111903599253911599863, 9.243636028279992175944653782056, 9.907111154658323456103239024973, 12.03413052079825843817815850242, 13.65135218437225710582321850655
