| 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
−10.06794545105895370894886736433, −9.582938423313310894927992893759, −8.682944066846789596164088504594, −7.35332365235382413653427795983, −7.05321959393413743925705119839, −5.46494391486788189832840844442, −4.86685053280178137874289080808, −3.58698784018896417907064872133, −1.85387923887005231103903729991, −0.17812309010728612900732288954,
1.27617103303613631443816099036, 2.97427671985590254136546223384, 3.46993666721800086059087491954, 5.06959721132716551507409450205, 6.11086264507494929058835039470, 7.19288963038443762580077838906, 8.433784483001456972133581187636, 9.170519940551655370284462414365, 10.34765503920438265851986254762, 10.87130547246286778684561590286
