| L(s) = 1 | − 18.4·2-s − 262.·3-s − 170.·4-s − 2.36e3·5-s + 4.84e3·6-s + 7.02e3·7-s + 1.26e4·8-s + 4.90e4·9-s + 4.36e4·10-s − 5.90e4·11-s + 4.46e4·12-s − 8.40e4·13-s − 1.29e5·14-s + 6.19e5·15-s − 1.46e5·16-s + 1.99e5·17-s − 9.06e5·18-s + 2.93e5·19-s + 4.02e5·20-s − 1.84e6·21-s + 1.09e6·22-s + 3.66e5·23-s − 3.30e6·24-s + 3.63e6·25-s + 1.55e6·26-s − 7.68e6·27-s − 1.19e6·28-s + ⋯ |
| L(s) = 1 | − 0.817·2-s − 1.86·3-s − 0.332·4-s − 1.69·5-s + 1.52·6-s + 1.10·7-s + 1.08·8-s + 2.49·9-s + 1.38·10-s − 1.21·11-s + 0.621·12-s − 0.815·13-s − 0.903·14-s + 3.15·15-s − 0.557·16-s + 0.578·17-s − 2.03·18-s + 0.516·19-s + 0.562·20-s − 2.06·21-s + 0.994·22-s + 0.273·23-s − 2.03·24-s + 1.85·25-s + 0.666·26-s − 2.78·27-s − 0.367·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 + 3.41e6T \) |
| good | 2 | \( 1 + 18.4T + 512T^{2} \) |
| 3 | \( 1 + 262.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.36e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.02e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.90e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 8.40e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.99e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.93e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.66e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.87e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.91e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.22e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.33e6T + 3.27e14T^{2} \) |
| 47 | \( 1 + 4.54e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.88e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.66e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.65e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.13e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.09e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 7.39e6T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.09e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.59e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.28e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.54e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.88508365349008470684414785192, −11.74098319407111090590262775795, −11.10391749287207714211140719381, −10.06632597344989331489419560507, −7.974653511941056439410392415140, −7.41048484416533410605183961359, −5.16207465907814167270559889600, −4.44687149929719658119756565382, −0.932072389055913516288814293645, 0,
0.932072389055913516288814293645, 4.44687149929719658119756565382, 5.16207465907814167270559889600, 7.41048484416533410605183961359, 7.974653511941056439410392415140, 10.06632597344989331489419560507, 11.10391749287207714211140719381, 11.74098319407111090590262775795, 12.88508365349008470684414785192
