| L(s) = 1 | − 81·3-s − 250.·5-s + 1.65e3·7-s + 6.56e3·9-s + 848.·11-s − 6.96e4·13-s + 2.02e4·15-s + 2.82e5·17-s + 4.70e5·19-s − 1.34e5·21-s − 2.41e6·23-s − 1.89e6·25-s − 5.31e5·27-s + 3.90e6·29-s + 3.99e6·31-s − 6.87e4·33-s − 4.14e5·35-s − 5.47e6·37-s + 5.64e6·39-s + 1.25e7·41-s + 1.12e7·43-s − 1.64e6·45-s − 2.89e7·47-s − 3.76e7·49-s − 2.28e7·51-s + 9.61e6·53-s − 2.12e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.179·5-s + 0.260·7-s + 0.333·9-s + 0.0174·11-s − 0.676·13-s + 0.103·15-s + 0.819·17-s + 0.829·19-s − 0.150·21-s − 1.80·23-s − 0.967·25-s − 0.192·27-s + 1.02·29-s + 0.776·31-s − 0.0100·33-s − 0.0466·35-s − 0.480·37-s + 0.390·39-s + 0.696·41-s + 0.502·43-s − 0.0596·45-s − 0.865·47-s − 0.932·49-s − 0.472·51-s + 0.167·53-s − 0.00313·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 81T \) |
| good | 5 | \( 1 + 250.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.65e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 848.T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.96e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.82e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.70e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.41e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.90e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.99e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.47e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.25e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.12e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.89e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.61e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.53e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.89e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.43e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.80e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.57e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.36e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.41e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.66e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.24e8T + 7.60e17T^{2} \) |
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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
−9.696472324076487343593148812596, −8.206436261856096419949445888432, −7.59947446316573385456033233961, −6.45252073075186216882322148654, −5.54220025260477530070923431989, −4.62614567111465600410455426588, −3.56675181638813698086904021299, −2.24747269121811233673702362298, −1.07094905498526588778991153290, 0,
1.07094905498526588778991153290, 2.24747269121811233673702362298, 3.56675181638813698086904021299, 4.62614567111465600410455426588, 5.54220025260477530070923431989, 6.45252073075186216882322148654, 7.59947446316573385456033233961, 8.206436261856096419949445888432, 9.696472324076487343593148812596
