| L(s) = 1 | + 4.84e6·3-s − 5.14e9·5-s − 2.72e11·7-s + 1.58e13·9-s − 9.16e13·11-s − 5.35e14·13-s − 2.49e16·15-s + 2.36e16·17-s + 8.85e16·19-s − 1.32e18·21-s + 1.77e18·23-s + 1.89e19·25-s + 4.00e19·27-s + 8.29e19·29-s + 3.33e19·31-s − 4.44e20·33-s + 1.40e21·35-s + 1.06e20·37-s − 2.59e21·39-s − 1.53e21·41-s + 1.62e22·43-s − 8.17e22·45-s − 1.37e22·47-s + 8.81e21·49-s + 1.14e23·51-s + 2.69e23·53-s + 4.71e23·55-s + ⋯ |
| L(s) = 1 | + 1.75·3-s − 1.88·5-s − 1.06·7-s + 2.08·9-s − 0.800·11-s − 0.489·13-s − 3.30·15-s + 0.579·17-s + 0.482·19-s − 1.87·21-s + 0.734·23-s + 2.54·25-s + 1.90·27-s + 1.50·29-s + 0.245·31-s − 1.40·33-s + 2.00·35-s + 0.0718·37-s − 0.860·39-s − 0.258·41-s + 1.43·43-s − 3.92·45-s − 0.368·47-s + 0.134·49-s + 1.01·51-s + 1.41·53-s + 1.50·55-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
| \(L(14)\) |
\(\approx\) |
\(2.263359339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.263359339\) |
| \(L(\frac{29}{2})\) |
|
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 | \( 1 - 4.84e6T + 7.62e12T^{2} \) |
| 5 | \( 1 + 5.14e9T + 7.45e18T^{2} \) |
| 7 | \( 1 + 2.72e11T + 6.57e22T^{2} \) |
| 11 | \( 1 + 9.16e13T + 1.31e28T^{2} \) |
| 13 | \( 1 + 5.35e14T + 1.19e30T^{2} \) |
| 17 | \( 1 - 2.36e16T + 1.66e33T^{2} \) |
| 19 | \( 1 - 8.85e16T + 3.36e34T^{2} \) |
| 23 | \( 1 - 1.77e18T + 5.84e36T^{2} \) |
| 29 | \( 1 - 8.29e19T + 3.05e39T^{2} \) |
| 31 | \( 1 - 3.33e19T + 1.84e40T^{2} \) |
| 37 | \( 1 - 1.06e20T + 2.19e42T^{2} \) |
| 41 | \( 1 + 1.53e21T + 3.50e43T^{2} \) |
| 43 | \( 1 - 1.62e22T + 1.26e44T^{2} \) |
| 47 | \( 1 + 1.37e22T + 1.40e45T^{2} \) |
| 53 | \( 1 - 2.69e23T + 3.59e46T^{2} \) |
| 59 | \( 1 + 2.79e22T + 6.50e47T^{2} \) |
| 61 | \( 1 + 9.04e23T + 1.59e48T^{2} \) |
| 67 | \( 1 - 6.95e23T + 2.01e49T^{2} \) |
| 71 | \( 1 + 1.63e25T + 9.63e49T^{2} \) |
| 73 | \( 1 + 9.30e24T + 2.04e50T^{2} \) |
| 79 | \( 1 + 2.80e24T + 1.72e51T^{2} \) |
| 83 | \( 1 + 1.27e25T + 6.53e51T^{2} \) |
| 89 | \( 1 - 1.64e26T + 4.30e52T^{2} \) |
| 97 | \( 1 - 9.73e26T + 4.39e53T^{2} \) |
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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
−13.14153549035429114224324258206, −12.12519759923829296111213682234, −10.25196806385322272628651314427, −8.903781848532066504508397745213, −7.87667739849360071815793125981, −7.14600432587731059987260510549, −4.47980551324392671075138300118, −3.31037505776386598837132894832, −2.80700369394313268045908865928, −0.69938293660749569340519667310,
0.69938293660749569340519667310, 2.80700369394313268045908865928, 3.31037505776386598837132894832, 4.47980551324392671075138300118, 7.14600432587731059987260510549, 7.87667739849360071815793125981, 8.903781848532066504508397745213, 10.25196806385322272628651314427, 12.12519759923829296111213682234, 13.14153549035429114224324258206
