| L(s) = 1 | + 2.48e5·3-s − 3.54e9·5-s + 2.27e11·7-s − 7.56e12·9-s + 1.01e13·11-s + 6.50e14·13-s − 8.81e14·15-s − 6.16e16·17-s − 2.66e17·19-s + 5.65e16·21-s + 2.43e18·23-s + 5.10e18·25-s − 3.77e18·27-s + 5.02e18·29-s − 2.01e19·31-s + 2.53e18·33-s − 8.05e20·35-s + 1.50e21·37-s + 1.61e20·39-s + 4.94e21·41-s − 1.43e22·43-s + 2.68e22·45-s − 4.74e22·47-s − 1.39e22·49-s − 1.53e22·51-s + 2.37e23·53-s − 3.61e22·55-s + ⋯ |
| L(s) = 1 | + 0.0900·3-s − 1.29·5-s + 0.887·7-s − 0.991·9-s + 0.0890·11-s + 0.595·13-s − 0.116·15-s − 1.51·17-s − 1.45·19-s + 0.0798·21-s + 1.00·23-s + 0.685·25-s − 0.179·27-s + 0.0908·29-s − 0.148·31-s + 0.00802·33-s − 1.15·35-s + 1.01·37-s + 0.0536·39-s + 0.834·41-s − 1.27·43-s + 1.28·45-s − 1.26·47-s − 0.213·49-s − 0.136·51-s + 1.25·53-s − 0.115·55-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
| \(L(14)\) |
\(\approx\) |
\(1.229592514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.229592514\) |
| \(L(\frac{29}{2})\) |
|
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 | \( 1 - 2.48e5T + 7.62e12T^{2} \) |
| 5 | \( 1 + 3.54e9T + 7.45e18T^{2} \) |
| 7 | \( 1 - 2.27e11T + 6.57e22T^{2} \) |
| 11 | \( 1 - 1.01e13T + 1.31e28T^{2} \) |
| 13 | \( 1 - 6.50e14T + 1.19e30T^{2} \) |
| 17 | \( 1 + 6.16e16T + 1.66e33T^{2} \) |
| 19 | \( 1 + 2.66e17T + 3.36e34T^{2} \) |
| 23 | \( 1 - 2.43e18T + 5.84e36T^{2} \) |
| 29 | \( 1 - 5.02e18T + 3.05e39T^{2} \) |
| 31 | \( 1 + 2.01e19T + 1.84e40T^{2} \) |
| 37 | \( 1 - 1.50e21T + 2.19e42T^{2} \) |
| 41 | \( 1 - 4.94e21T + 3.50e43T^{2} \) |
| 43 | \( 1 + 1.43e22T + 1.26e44T^{2} \) |
| 47 | \( 1 + 4.74e22T + 1.40e45T^{2} \) |
| 53 | \( 1 - 2.37e23T + 3.59e46T^{2} \) |
| 59 | \( 1 - 1.40e24T + 6.50e47T^{2} \) |
| 61 | \( 1 + 1.05e24T + 1.59e48T^{2} \) |
| 67 | \( 1 - 7.87e24T + 2.01e49T^{2} \) |
| 71 | \( 1 - 1.00e25T + 9.63e49T^{2} \) |
| 73 | \( 1 - 2.10e25T + 2.04e50T^{2} \) |
| 79 | \( 1 + 5.00e25T + 1.72e51T^{2} \) |
| 83 | \( 1 - 1.13e26T + 6.53e51T^{2} \) |
| 89 | \( 1 - 1.60e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 2.68e26T + 4.39e53T^{2} \) |
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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
−13.06235111946906377698459899186, −11.46186414060994266142064459640, −11.05097098764588178388013438414, −8.750118219323210989983526360573, −8.095224266438183877299494351856, −6.57073315211889655529592969267, −4.82995365379324659246786382869, −3.75495169619207074709089139478, −2.24755530357198709186379764393, −0.55369091138267194760748460350,
0.55369091138267194760748460350, 2.24755530357198709186379764393, 3.75495169619207074709089139478, 4.82995365379324659246786382869, 6.57073315211889655529592969267, 8.095224266438183877299494351856, 8.750118219323210989983526360573, 11.05097098764588178388013438414, 11.46186414060994266142064459640, 13.06235111946906377698459899186
