| L(s) = 1 | + i·2-s − 4-s − 1.73i·5-s − 1.26i·7-s − i·8-s + 1.73·10-s + 1.26i·11-s + 1.26·14-s + 16-s + 5.19·17-s + 4.73i·19-s + 1.73i·20-s − 1.26·22-s − 8.19·23-s + 2.00·25-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.774i·5-s − 0.479i·7-s − 0.353i·8-s + 0.547·10-s + 0.382i·11-s + 0.338·14-s + 0.250·16-s + 1.26·17-s + 1.08i·19-s + 0.387i·20-s − 0.270·22-s − 1.70·23-s + 0.400·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.700817038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.700817038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 7 | \( 1 + 1.26iT - 7T^{2} \) |
| 11 | \( 1 - 1.26iT - 11T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 4.73iT - 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 9.46iT - 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 + 6.46iT - 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 + 4.73iT - 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + 13.8iT - 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 7.26iT - 67T^{2} \) |
| 71 | \( 1 - 2.19iT - 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 + 5.66iT - 83T^{2} \) |
| 89 | \( 1 - 9.46iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.528197878448980449414463757416, −8.023083020621778224621612373009, −7.32802233318404623955482718754, −6.51413863725197972569277820583, −5.61531728575718966664116971954, −5.07554022201380351734413285253, −4.11283763399458967427802181847, −3.46794654278528226964588061284, −1.88215650738788830666285664808, −0.76342137354852615322798371255,
0.855595296510717856636467497669, 2.27927250007156010545737945339, 2.88026134974295139236782063997, 3.76992010708974854253397926124, 4.66487284234912939850531083849, 5.71791356953175737763660379785, 6.23219618402024318917055737516, 7.29730763584879703496422085345, 8.023147022633971543702964668515, 8.727927020371192527495451909402
