| L(s) = 1 | + 39.8·2-s + 51.5·3-s + 1.07e3·4-s + 643.·5-s + 2.05e3·6-s − 8.54e3·7-s + 2.25e4·8-s − 1.70e4·9-s + 2.56e4·10-s + 1.04e4·11-s + 5.55e4·12-s − 4.87e4·13-s − 3.40e5·14-s + 3.31e4·15-s + 3.48e5·16-s − 6.79e5·18-s − 1.57e5·19-s + 6.93e5·20-s − 4.39e5·21-s + 4.18e5·22-s + 2.48e6·23-s + 1.16e6·24-s − 1.53e6·25-s − 1.94e6·26-s − 1.89e6·27-s − 9.20e6·28-s − 3.86e6·29-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.367·3-s + 2.10·4-s + 0.460·5-s + 0.646·6-s − 1.34·7-s + 1.94·8-s − 0.865·9-s + 0.811·10-s + 0.216·11-s + 0.773·12-s − 0.473·13-s − 2.36·14-s + 0.168·15-s + 1.32·16-s − 1.52·18-s − 0.276·19-s + 0.969·20-s − 0.493·21-s + 0.381·22-s + 1.85·23-s + 0.715·24-s − 0.788·25-s − 0.834·26-s − 0.684·27-s − 2.83·28-s − 1.01·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 - 39.8T + 512T^{2} \) |
| 3 | \( 1 - 51.5T + 1.96e4T^{2} \) |
| 5 | \( 1 - 643.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.54e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.04e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 4.87e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 1.57e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.48e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.86e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.94e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.18e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.89e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.91e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.40e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.91e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.82e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.75e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 9.64e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.59e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 9.76e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 7.15e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.09e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.08e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.19e8T + 7.60e17T^{2} \) |
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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
−9.821908691944808390232059499471, −8.929322657990816256198995130927, −7.35951120730172431759058660658, −6.43581565520484081370549467376, −5.76018589726744945450639171400, −4.76869626572163207961628701343, −3.42254493271396193156250789837, −2.99907868892653629230581019911, −1.90913970945843584311366892301, 0,
1.90913970945843584311366892301, 2.99907868892653629230581019911, 3.42254493271396193156250789837, 4.76869626572163207961628701343, 5.76018589726744945450639171400, 6.43581565520484081370549467376, 7.35951120730172431759058660658, 8.929322657990816256198995130927, 9.821908691944808390232059499471
