| L(s) = 1 | + 0.473i·2-s + 1.77·4-s − 2.56i·7-s + 1.78i·8-s + 6.16·11-s + 2.13i·13-s + 1.21·14-s + 2.70·16-s + 3.16i·17-s − 0.356·19-s + 2.91i·22-s + 4.21i·23-s − 1.00·26-s − 4.55i·28-s + 1.68·29-s + ⋯ |
| L(s) = 1 | + 0.334i·2-s + 0.888·4-s − 0.968i·7-s + 0.631i·8-s + 1.85·11-s + 0.591i·13-s + 0.324·14-s + 0.676·16-s + 0.768i·17-s − 0.0817·19-s + 0.622i·22-s + 0.878i·23-s − 0.197·26-s − 0.860i·28-s + 0.313·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.550435690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.550435690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.473iT - 2T^{2} \) |
| 7 | \( 1 + 2.56iT - 7T^{2} \) |
| 11 | \( 1 - 6.16T + 11T^{2} \) |
| 13 | \( 1 - 2.13iT - 13T^{2} \) |
| 17 | \( 1 - 3.16iT - 17T^{2} \) |
| 19 | \( 1 + 0.356T + 19T^{2} \) |
| 23 | \( 1 - 4.21iT - 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 + 8.25T + 31T^{2} \) |
| 37 | \( 1 + 3.63iT - 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 + 7.67iT - 43T^{2} \) |
| 47 | \( 1 + 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 9.43iT - 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 0.0109T + 61T^{2} \) |
| 67 | \( 1 + 0.982iT - 67T^{2} \) |
| 71 | \( 1 - 6.43T + 71T^{2} \) |
| 73 | \( 1 - 6.61iT - 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 - 10.4iT - 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + 7.20iT - 97T^{2} \) |
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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
−9.127093752796057772839380768926, −8.398761455374523061549596255183, −7.24818869183224878327248253897, −7.03287972951821383446452266285, −6.22013711920518566876644825152, −5.40545762308668193547422677829, −3.96553662604386498373347554424, −3.71280667756604151921581475353, −2.08524687645438160320750296584, −1.22926269716302564167703473118,
1.09289610562692701412847447362, 2.18785395626822785004816997589, 3.05776874434996838763539315916, 3.96440540055782343535008519928, 5.12481600964179118968203671770, 6.13671762466848985674179968323, 6.57415887073536302577548114530, 7.45129189836696774590591774795, 8.398433586052818670464212910807, 9.231005685695923357649224537896
