| L(s) = 1 | − i·2-s − 4-s + 2i·7-s + i·8-s − 3·11-s − 5i·13-s + 2·14-s + 16-s + 3i·17-s + 4·19-s + 3i·22-s − 9i·23-s − 5·26-s − 2i·28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.755i·7-s + 0.353i·8-s − 0.904·11-s − 1.38i·13-s + 0.534·14-s + 0.250·16-s + 0.727i·17-s + 0.917·19-s + 0.639i·22-s − 1.87i·23-s − 0.980·26-s − 0.377i·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.202156476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.202156476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.495157831553610679814199769671, −8.397653175916324563837969921205, −8.146390955941154337500793267583, −6.92065656097945310407842743974, −5.64086534872231543205755474175, −5.28761980035629810492378082530, −4.02848678937883208122259448399, −2.93583972209967448686707042237, −2.22970871807828956655362370397, −0.53834880395933276273702940229,
1.28861144535864305863726083217, 2.91946066844286205900365904418, 4.04565519395004100741305244654, 4.89340176129836874320029489098, 5.71320591401434983249205438488, 6.77783051042951068768569766255, 7.41641084694036046840800316640, 7.980300497392279542860719910665, 9.160288088543018681544662586376, 9.647668444572553716119116250019
