L(s) = 1 | + (1.16 + 2.01i)5-s + (1.25 − 2.33i)7-s + (−2.17 − 1.25i)11-s + (0.244 − 0.141i)13-s + 2.01·17-s − 2.93i·19-s + (−4.32 + 2.49i)23-s + (−0.199 + 0.345i)25-s + (−4.02 − 2.32i)29-s + (8.91 − 5.14i)31-s + (6.14 − 0.191i)35-s + 3.93·37-s + (3.44 + 5.97i)41-s + (5.66 − 9.80i)43-s + (1.84 − 3.19i)47-s + ⋯ |
L(s) = 1 | + (0.519 + 0.899i)5-s + (0.472 − 0.881i)7-s + (−0.656 − 0.379i)11-s + (0.0678 − 0.0391i)13-s + 0.487·17-s − 0.674i·19-s + (−0.901 + 0.520i)23-s + (−0.0398 + 0.0690i)25-s + (−0.747 − 0.431i)29-s + (1.60 − 0.924i)31-s + (1.03 − 0.0324i)35-s + 0.646·37-s + (0.538 + 0.932i)41-s + (0.863 − 1.49i)43-s + (0.268 − 0.465i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.952584159\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.952584159\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.25 + 2.33i)T \) |
good | 5 | \( 1 + (-1.16 - 2.01i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.17 + 1.25i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.244 + 0.141i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2.01T + 17T^{2} \) |
| 19 | \( 1 + 2.93iT - 19T^{2} \) |
| 23 | \( 1 + (4.32 - 2.49i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.02 + 2.32i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-8.91 + 5.14i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 + (-3.44 - 5.97i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.66 + 9.80i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.84 + 3.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.845iT - 53T^{2} \) |
| 59 | \( 1 + (7.27 + 12.6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (9.59 + 5.54i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.59 - 9.69i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.67iT - 71T^{2} \) |
| 73 | \( 1 - 6.37iT - 73T^{2} \) |
| 79 | \( 1 + (3.71 - 6.43i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.73 + 8.19i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 5.12T + 89T^{2} \) |
| 97 | \( 1 + (-4.49 - 2.59i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.447890994953309095184246621197, −7.79234977635195537025501991966, −7.22556311711218922624509085636, −6.31133369473765169096566420508, −5.72980500130508683618650701586, −4.71242929754887686993788686582, −3.86430358628545876669465095647, −2.89075252859563878220635246282, −2.06518008911393208078743416216, −0.66103357253439009931195036424,
1.18586478387106258754397836272, 2.10588565988221410535088654466, 3.03422770921252073240316584478, 4.39553063879215040186696420696, 4.95328121602481295162593507150, 5.78205951813814628589636797032, 6.23653605702765807333216621939, 7.65521412168784778621118673836, 7.991757515790777428664138256993, 8.964051808485411457370567364175