| L(s) = 1 | + 81·3-s + 1.76e3·5-s − 255.·7-s + 6.56e3·9-s − 6.11e4·11-s − 5.08e4·13-s + 1.43e5·15-s + 3.89e5·17-s − 8.66e5·19-s − 2.06e4·21-s + 9.81e5·23-s + 1.16e6·25-s + 5.31e5·27-s + 2.03e6·29-s − 2.77e6·31-s − 4.94e6·33-s − 4.50e5·35-s + 3.50e6·37-s − 4.11e6·39-s + 5.84e6·41-s − 2.78e7·43-s + 1.15e7·45-s − 1.99e7·47-s − 4.02e7·49-s + 3.15e7·51-s + 4.12e7·53-s − 1.07e8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.26·5-s − 0.0401·7-s + 0.333·9-s − 1.25·11-s − 0.493·13-s + 0.729·15-s + 1.13·17-s − 1.52·19-s − 0.0231·21-s + 0.731·23-s + 0.596·25-s + 0.192·27-s + 0.535·29-s − 0.539·31-s − 0.726·33-s − 0.0507·35-s + 0.307·37-s − 0.285·39-s + 0.322·41-s − 1.24·43-s + 0.421·45-s − 0.597·47-s − 0.998·49-s + 0.653·51-s + 0.717·53-s − 1.59·55-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 | \( 1 - 1.76e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 255.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.11e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.08e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.89e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.66e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.81e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.03e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.77e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.50e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 5.84e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.78e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.99e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.12e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.84e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.02e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.49e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.07e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.01e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.25e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.33e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.45e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.41e8T + 7.60e17T^{2} \) |
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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
−9.474557818298349822774812170165, −8.470366630354100020851777791431, −7.61639885773906046054871796713, −6.49364205886708644734182578073, −5.53252303139876203209538893902, −4.66964844170006484196477152912, −3.15292350524527534683077638777, −2.36251968585045367604155709884, −1.47271062461248909238258883567, 0,
1.47271062461248909238258883567, 2.36251968585045367604155709884, 3.15292350524527534683077638777, 4.66964844170006484196477152912, 5.53252303139876203209538893902, 6.49364205886708644734182578073, 7.61639885773906046054871796713, 8.470366630354100020851777791431, 9.474557818298349822774812170165
