L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 14.9i·5-s − 18.5·7-s + 22.6i·8-s + 42.3·10-s − 207. i·11-s − 62.8·13-s + 52.3i·14-s + 64.0·16-s + 187. i·17-s + 462.·19-s − 119. i·20-s − 587.·22-s − 455. i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.598i·5-s − 0.377·7-s + 0.353i·8-s + 0.423·10-s − 1.71i·11-s − 0.372·13-s + 0.267i·14-s + 0.250·16-s + 0.648i·17-s + 1.28·19-s − 0.299i·20-s − 1.21·22-s − 0.861i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5950210203\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5950210203\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 18.5T \) |
good | 5 | \( 1 - 14.9iT - 625T^{2} \) |
| 11 | \( 1 + 207. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 62.8T + 2.85e4T^{2} \) |
| 17 | \( 1 - 187. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 462.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 455. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.18e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.51e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.09e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.99e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.08e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.65e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.86e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.12e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.22e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.23e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.62e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 6.23e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.40e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 6.08e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.33e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 6.07e3T + 8.85e7T^{2} \) |
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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
−10.87130547246286778684561590286, −10.34765503920438265851986254762, −9.170519940551655370284462414365, −8.433784483001456972133581187636, −7.19288963038443762580077838906, −6.11086264507494929058835039470, −5.06959721132716551507409450205, −3.46993666721800086059087491954, −2.97427671985590254136546223384, −1.27617103303613631443816099036,
0.17812309010728612900732288954, 1.85387923887005231103903729991, 3.58698784018896417907064872133, 4.86685053280178137874289080808, 5.46494391486788189832840844442, 7.05321959393413743925705119839, 7.35332365235382413653427795983, 8.682944066846789596164088504594, 9.582938423313310894927992893759, 10.06794545105895370894886736433