| L(s) = 1 | + (−1.22 + 1.22i)3-s − 2.44·5-s − i·7-s − 2.99i·9-s + 2.44i·13-s + (2.99 − 2.99i)15-s − 6i·17-s + 7.34·19-s + (1.22 + 1.22i)21-s + 0.999·25-s + (3.67 + 3.67i)27-s − 4.89·29-s + 8i·31-s + 2.44i·35-s + 4.89i·37-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)3-s − 1.09·5-s − 0.377i·7-s − 0.999i·9-s + 0.679i·13-s + (0.774 − 0.774i)15-s − 1.45i·17-s + 1.68·19-s + (0.267 + 0.267i)21-s + 0.199·25-s + (0.707 + 0.707i)27-s − 0.909·29-s + 1.43i·31-s + 0.414i·35-s + 0.805i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7682334052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7682334052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 7.34T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4.89T + 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + 4.89T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 9.79T + 53T^{2} \) |
| 59 | \( 1 - 2.44iT - 59T^{2} \) |
| 61 | \( 1 + 2.44iT - 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 8iT - 79T^{2} \) |
| 83 | \( 1 - 7.34iT - 83T^{2} \) |
| 89 | \( 1 - 18iT - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.672886102732292149639893300829, −9.313482303317624281736049442096, −8.132115582604733399521514833343, −7.25682762970968026041068963923, −6.69227441613206067540661816872, −5.36938608877597807671980636344, −4.77418507018176416283082275476, −3.83774917698097407002216912548, −3.08871469462771919111532343046, −0.984576197356245551386223654532,
0.45543606542336692927676059052, 1.90300953118654380244777770384, 3.30946589493670573847160165927, 4.25447359351113661426920067833, 5.48542527222416405323664276701, 5.92810601130871390578055574164, 7.15594345008757724437062849574, 7.72870553572076302077829856456, 8.276202628913869584887930693442, 9.388940324415910139418779944798
