| L(s) = 1 | + 9.17·2-s − 427.·4-s − 625·5-s + 2.40e3·7-s − 8.62e3·8-s − 5.73e3·10-s − 3.50e4·11-s − 7.74e4·13-s + 2.20e4·14-s + 1.40e5·16-s + 2.29e5·17-s + 1.64e4·19-s + 2.67e5·20-s − 3.21e5·22-s + 2.57e6·23-s + 3.90e5·25-s − 7.09e5·26-s − 1.02e6·28-s + 6.62e6·29-s − 8.17e6·31-s + 5.69e6·32-s + 2.10e6·34-s − 1.50e6·35-s + 9.70e6·37-s + 1.50e5·38-s + 5.38e6·40-s − 2.98e7·41-s + ⋯ |
| L(s) = 1 | + 0.405·2-s − 0.835·4-s − 0.447·5-s + 0.377·7-s − 0.744·8-s − 0.181·10-s − 0.722·11-s − 0.751·13-s + 0.153·14-s + 0.534·16-s + 0.667·17-s + 0.0289·19-s + 0.373·20-s − 0.292·22-s + 1.91·23-s + 0.200·25-s − 0.304·26-s − 0.315·28-s + 1.74·29-s − 1.58·31-s + 0.960·32-s + 0.270·34-s − 0.169·35-s + 0.851·37-s + 0.0117·38-s + 0.332·40-s − 1.64·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 - 9.17T + 512T^{2} \) |
| 11 | \( 1 + 3.50e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.74e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.29e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.64e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.57e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.17e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.70e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.98e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.95e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.93e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.74e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.24e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.23e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.74e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.63e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.09e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.65e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 9.43e6T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.64e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.20e9T + 7.60e17T^{2} \) |
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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
−9.618058831122653312703488003445, −8.621257696450261850908629465602, −7.84518109545568595050095107099, −6.77745056099952540050459675318, −5.22121719267353138036361413613, −4.92858317129229669751979002112, −3.64228408740342383077821691498, −2.70995876456666616128621255726, −1.05421820987400243223177036575, 0,
1.05421820987400243223177036575, 2.70995876456666616128621255726, 3.64228408740342383077821691498, 4.92858317129229669751979002112, 5.22121719267353138036361413613, 6.77745056099952540050459675318, 7.84518109545568595050095107099, 8.621257696450261850908629465602, 9.618058831122653312703488003445
