| L(s) = 1 | + 6.62·2-s + 11.8·4-s + 85.2·5-s − 103.·7-s − 133.·8-s + 564.·10-s − 579.·11-s + 435.·13-s − 682.·14-s − 1.26e3·16-s − 424.·17-s + 1.54e3·19-s + 1.01e3·20-s − 3.84e3·22-s + 4.16e3·23-s + 4.13e3·25-s + 2.88e3·26-s − 1.22e3·28-s + 8.41e3·29-s + 7.07e3·31-s − 4.10e3·32-s − 2.81e3·34-s − 8.77e3·35-s + 1.24e4·37-s + 1.02e4·38-s − 1.13e4·40-s + 2.42e3·41-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.371·4-s + 1.52·5-s − 0.794·7-s − 0.735·8-s + 1.78·10-s − 1.44·11-s + 0.714·13-s − 0.930·14-s − 1.23·16-s − 0.355·17-s + 0.979·19-s + 0.566·20-s − 1.69·22-s + 1.64·23-s + 1.32·25-s + 0.837·26-s − 0.295·28-s + 1.85·29-s + 1.32·31-s − 0.708·32-s − 0.416·34-s − 1.21·35-s + 1.49·37-s + 1.14·38-s − 1.12·40-s + 0.224·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.484901150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.484901150\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 - 6.62T + 32T^{2} \) |
| 5 | \( 1 - 85.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 103.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 579.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 435.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 424.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.54e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.16e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.24e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.42e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.43e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.37e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 5.56e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.78e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.76e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.66e5T + 8.58e9T^{2} \) |
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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
−10.03781944994428027893651529459, −9.359744699891372416302397502701, −8.384812214163401303219827402307, −6.83356834528086383708433659527, −6.11900064039164373653160975506, −5.38287617136266585376713022789, −4.62690989170285948560482181988, −3.01902021429493018116361830757, −2.63969618660858512975277542879, −0.892604449198739950120370587694,
0.892604449198739950120370587694, 2.63969618660858512975277542879, 3.01902021429493018116361830757, 4.62690989170285948560482181988, 5.38287617136266585376713022789, 6.11900064039164373653160975506, 6.83356834528086383708433659527, 8.384812214163401303219827402307, 9.359744699891372416302397502701, 10.03781944994428027893651529459
