| L(s) = 1 | − i·3-s + (1 − 2i)5-s − i·7-s − 9-s + 2·11-s + 2i·13-s + (−2 − i)15-s − 6·19-s − 21-s + 8i·23-s + (−3 − 4i)25-s + i·27-s − 6·29-s − 10·31-s − 2i·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.447 − 0.894i)5-s − 0.377i·7-s − 0.333·9-s + 0.603·11-s + 0.554i·13-s + (−0.516 − 0.258i)15-s − 1.37·19-s − 0.218·21-s + 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.192i·27-s − 1.11·29-s − 1.79·31-s − 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5551001197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5551001197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.274512137559448455460535201643, −7.31629850307708500964429993834, −6.85000642018443407104843059337, −5.79431844707338327410518512145, −5.37476953136403275462481248555, −4.18834287360190066692406421460, −3.63499789754054518610993517307, −2.01100176092952639044898944376, −1.59462033990013488894225854547, −0.15157138060999760872668201614,
1.78303011523515690794133173816, 2.69092832449966498426826711707, 3.51718245214933408380273794475, 4.37898966101054919940999309713, 5.29514266980052317167154322762, 6.17629148876581429129953919448, 6.57160460943585065057136817172, 7.55628711068129622695664873114, 8.448031672138903000288655699172, 9.077982012850391101525928114207
