| L(s) = 1 | + (−1.48 − 1.67i)5-s + i·7-s − 0.387·11-s + 2.96i·13-s − 3.35i·17-s + 2.96·19-s − 0.962i·23-s + (−0.612 + 4.96i)25-s + 1.22·29-s + 2.96·31-s + (1.67 − 1.48i)35-s + 5.92i·37-s − 1.03·41-s − 10.7i·43-s − 3.22i·47-s + ⋯ |
| L(s) = 1 | + (−0.662 − 0.749i)5-s + 0.377i·7-s − 0.116·11-s + 0.821i·13-s − 0.812i·17-s + 0.679·19-s − 0.200i·23-s + (−0.122 + 0.992i)25-s + 0.227·29-s + 0.532·31-s + (0.283 − 0.250i)35-s + 0.974i·37-s − 0.162·41-s − 1.63i·43-s − 0.470i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.410905977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410905977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.48 + 1.67i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 0.387T + 11T^{2} \) |
| 13 | \( 1 - 2.96iT - 13T^{2} \) |
| 17 | \( 1 + 3.35iT - 17T^{2} \) |
| 19 | \( 1 - 2.96T + 19T^{2} \) |
| 23 | \( 1 + 0.962iT - 23T^{2} \) |
| 29 | \( 1 - 1.22T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 - 5.92iT - 37T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 + 10.7iT - 43T^{2} \) |
| 47 | \( 1 + 3.22iT - 47T^{2} \) |
| 53 | \( 1 + 5.66iT - 53T^{2} \) |
| 59 | \( 1 - 3.22T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.53T + 71T^{2} \) |
| 73 | \( 1 + 6.18iT - 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 + 7.73iT - 97T^{2} \) |
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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
−8.730604011946123710626834054047, −8.195852827051917876547229334321, −7.27338110184000673918904854565, −6.64698225085587035244092183584, −5.47319099502635971253798709573, −4.90598111915473289064967176161, −4.05350306090576101591252575665, −3.10799741175284006871333470341, −1.92787604463842907377696639753, −0.61722806396296162713236386538,
0.934014657597847936100595320934, 2.46544849542397634251459926215, 3.35770122588124166342951182956, 4.05532566677377293450828484909, 5.06484050231403991468046612239, 6.03335941324992163157957194901, 6.74109606981050425460159713956, 7.69664127292486778614630581988, 7.945267296020072153243906137263, 8.949033358425197978095671490485
