| L(s) = 1 | − 3.31e6·3-s − 1.51e9·5-s − 7.50e10·7-s + 3.35e12·9-s + 4.82e13·11-s − 1.67e15·13-s + 5.02e15·15-s − 3.32e16·17-s − 9.72e16·19-s + 2.48e17·21-s − 2.52e18·23-s − 5.15e18·25-s + 1.41e19·27-s − 7.47e19·29-s − 1.99e20·31-s − 1.59e20·33-s + 1.13e20·35-s − 1.37e21·37-s + 5.54e21·39-s − 1.75e21·41-s + 1.71e22·43-s − 5.08e21·45-s + 1.00e22·47-s − 6.00e22·49-s + 1.10e23·51-s − 2.26e23·53-s − 7.31e22·55-s + ⋯ |
| L(s) = 1 | − 1.20·3-s − 0.555·5-s − 0.292·7-s + 0.440·9-s + 0.421·11-s − 1.53·13-s + 0.666·15-s − 0.813·17-s − 0.530·19-s + 0.351·21-s − 1.04·23-s − 0.691·25-s + 0.671·27-s − 1.35·29-s − 1.47·31-s − 0.505·33-s + 0.162·35-s − 0.927·37-s + 1.83·39-s − 0.296·41-s + 1.52·43-s − 0.244·45-s + 0.267·47-s − 0.914·49-s + 0.976·51-s − 1.19·53-s − 0.234·55-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
| \(L(14)\) |
\(\approx\) |
\(0.09462632947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09462632947\) |
| \(L(\frac{29}{2})\) |
|
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 | \( 1 + 3.31e6T + 7.62e12T^{2} \) |
| 5 | \( 1 + 1.51e9T + 7.45e18T^{2} \) |
| 7 | \( 1 + 7.50e10T + 6.57e22T^{2} \) |
| 11 | \( 1 - 4.82e13T + 1.31e28T^{2} \) |
| 13 | \( 1 + 1.67e15T + 1.19e30T^{2} \) |
| 17 | \( 1 + 3.32e16T + 1.66e33T^{2} \) |
| 19 | \( 1 + 9.72e16T + 3.36e34T^{2} \) |
| 23 | \( 1 + 2.52e18T + 5.84e36T^{2} \) |
| 29 | \( 1 + 7.47e19T + 3.05e39T^{2} \) |
| 31 | \( 1 + 1.99e20T + 1.84e40T^{2} \) |
| 37 | \( 1 + 1.37e21T + 2.19e42T^{2} \) |
| 41 | \( 1 + 1.75e21T + 3.50e43T^{2} \) |
| 43 | \( 1 - 1.71e22T + 1.26e44T^{2} \) |
| 47 | \( 1 - 1.00e22T + 1.40e45T^{2} \) |
| 53 | \( 1 + 2.26e23T + 3.59e46T^{2} \) |
| 59 | \( 1 + 1.00e24T + 6.50e47T^{2} \) |
| 61 | \( 1 - 1.34e24T + 1.59e48T^{2} \) |
| 67 | \( 1 - 5.33e24T + 2.01e49T^{2} \) |
| 71 | \( 1 + 7.13e24T + 9.63e49T^{2} \) |
| 73 | \( 1 - 6.77e24T + 2.04e50T^{2} \) |
| 79 | \( 1 - 2.79e25T + 1.72e51T^{2} \) |
| 83 | \( 1 - 9.54e25T + 6.53e51T^{2} \) |
| 89 | \( 1 - 3.43e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 4.35e25T + 4.39e53T^{2} \) |
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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
−12.68555227450607779791960145295, −11.80708479664019110807298842541, −10.78585583532101940849067545807, −9.364386869362460506478533345096, −7.57747903000227396168145152819, −6.37163098535012273445498738032, −5.14138344108970207485010833905, −3.89518578927680416759848005635, −2.05276621156574229601378710517, −0.15910852646769645689039531470,
0.15910852646769645689039531470, 2.05276621156574229601378710517, 3.89518578927680416759848005635, 5.14138344108970207485010833905, 6.37163098535012273445498738032, 7.57747903000227396168145152819, 9.364386869362460506478533345096, 10.78585583532101940849067545807, 11.80708479664019110807298842541, 12.68555227450607779791960145295
