| 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
| \(L(1)\) |
\(\approx\) |
\(1.682823568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682823568\) |
| \(L(\frac{3}{2})\) |
|
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
−9.610142161869288624425922836330, −8.758243441785760467368790754919, −8.073459110703098147115651480086, −6.78409240930332923606902787916, −6.42898947484056013059860306590, −4.88743157405223783151079423933, −4.23461448106154454574024456109, −3.39790647796439873175836056324, −2.07318364568298566846971972377, −0.946975320063420592032284861489,
1.05915460607740131976311934600, 2.77065246227671345359376608056, 3.70958171349738078958363352498, 4.97700878286273400553039037136, 5.63794483798835075050925748625, 6.34685536866354225943388650549, 7.51632023558408280520421044373, 7.86940278203741857465476551266, 9.175842006182293757196212232098, 9.306379986505546483557223769992
