| L(s) = 1 | − i·3-s − 3.82i·5-s − 3.82·7-s + 2·9-s − 4.82i·11-s − i·13-s − 3.82·15-s + 6.65·17-s − 4i·19-s + 3.82i·21-s + 3.17·23-s − 9.65·25-s − 5i·27-s + 3.17i·29-s − 4.82·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.71i·5-s − 1.44·7-s + 0.666·9-s − 1.45i·11-s − 0.277i·13-s − 0.988·15-s + 1.61·17-s − 0.917i·19-s + 0.835i·21-s + 0.661·23-s − 1.93·25-s − 0.962i·27-s + 0.588i·29-s − 0.840·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.319562917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319562917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 + 3.82iT - 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 + 4.82iT - 11T^{2} \) |
| 17 | \( 1 - 6.65T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 3.17iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 3.82iT - 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 3iT - 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 3.17iT - 53T^{2} \) |
| 59 | \( 1 - 5.17iT - 59T^{2} \) |
| 61 | \( 1 - 10.8iT - 61T^{2} \) |
| 67 | \( 1 - 3.65iT - 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 5.17T + 73T^{2} \) |
| 79 | \( 1 + 7.65T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.942805314875948784480640973249, −8.275256935335078245836838577897, −7.43027805763581241714681152038, −6.49555083645606059862516911874, −5.69783543697088954948748130588, −4.99427480882994889157505820577, −3.78538856769736039610229217096, −2.97143567693439979278116289642, −1.25721173096117824178596601706, −0.55881012528911345339383383343,
1.91561348347838985988382611515, 3.23943789179362477024018036808, 3.52035506803693140023263619047, 4.68505225335158643445240062285, 5.90540722223318409395320809984, 6.72238219399123496399359574048, 7.14133337642790199125666238086, 7.941118030439146049134725950432, 9.563602948454538897134971442903, 9.857937462364912010351302590808
