| L(s) = 1 | + 16.8·2-s − 229.·4-s + 1.10e3·5-s − 5.16e3·7-s − 1.24e4·8-s + 1.85e4·10-s − 4.45e4·11-s + 6.96e4·13-s − 8.68e4·14-s − 9.20e4·16-s − 8.35e4·17-s − 1.70e5·19-s − 2.53e5·20-s − 7.48e5·22-s + 2.00e6·23-s − 7.35e5·25-s + 1.17e6·26-s + 1.18e6·28-s − 1.55e5·29-s + 4.27e6·31-s + 4.83e6·32-s − 1.40e6·34-s − 5.69e6·35-s + 1.51e7·37-s − 2.86e6·38-s − 1.37e7·40-s − 1.59e7·41-s + ⋯ |
| L(s) = 1 | + 0.742·2-s − 0.447·4-s + 0.789·5-s − 0.812·7-s − 1.07·8-s + 0.586·10-s − 0.917·11-s + 0.676·13-s − 0.604·14-s − 0.351·16-s − 0.242·17-s − 0.299·19-s − 0.353·20-s − 0.681·22-s + 1.49·23-s − 0.376·25-s + 0.502·26-s + 0.364·28-s − 0.0408·29-s + 0.831·31-s + 0.814·32-s − 0.180·34-s − 0.641·35-s + 1.32·37-s − 0.222·38-s − 0.849·40-s − 0.880·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.304321078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.304321078\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + 8.35e4T \) |
| good | 2 | \( 1 - 16.8T + 512T^{2} \) |
| 5 | \( 1 - 1.10e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.16e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.45e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.96e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 1.70e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.00e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.55e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.51e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.59e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.49e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.36e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.50e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.94e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.27e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.73e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.88e6T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.32e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.38e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.74e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.82e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.03e9T + 7.60e17T^{2} \) |
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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
−11.35849650327910499037382019942, −10.12950997682980123356770619614, −9.340630914171245183526498464652, −8.301291685762194571738211176638, −6.64935699880399344026918222797, −5.78543765103365999281793221972, −4.82260643699283924681324708824, −3.49933031496656201087541968996, −2.47169009356354744169621501081, −0.67856141200999475378795106016,
0.67856141200999475378795106016, 2.47169009356354744169621501081, 3.49933031496656201087541968996, 4.82260643699283924681324708824, 5.78543765103365999281793221972, 6.64935699880399344026918222797, 8.301291685762194571738211176638, 9.340630914171245183526498464652, 10.12950997682980123356770619614, 11.35849650327910499037382019942
