L(s) = 1 | + (0.0868 − 0.150i)5-s + (2.60 − 0.486i)7-s + (3.25 − 1.87i)11-s + (−3.54 − 2.04i)13-s + 6.00·17-s + 6.26i·19-s + (−6.37 − 3.68i)23-s + (2.48 + 4.30i)25-s + (8.58 − 4.95i)29-s + (−3.76 − 2.17i)31-s + (0.152 − 0.433i)35-s + 7.23·37-s + (0.489 − 0.848i)41-s + (−0.0468 − 0.0811i)43-s + (−1.86 − 3.22i)47-s + ⋯ |
L(s) = 1 | + (0.0388 − 0.0672i)5-s + (0.982 − 0.184i)7-s + (0.980 − 0.566i)11-s + (−0.982 − 0.567i)13-s + 1.45·17-s + 1.43i·19-s + (−1.32 − 0.767i)23-s + (0.496 + 0.860i)25-s + (1.59 − 0.920i)29-s + (−0.676 − 0.390i)31-s + (0.0257 − 0.0732i)35-s + 1.18·37-s + (0.0765 − 0.132i)41-s + (−0.00714 − 0.0123i)43-s + (−0.271 − 0.470i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.247294431\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.247294431\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.60 + 0.486i)T \) |
good | 5 | \( 1 + (-0.0868 + 0.150i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.25 + 1.87i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.54 + 2.04i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6.00T + 17T^{2} \) |
| 19 | \( 1 - 6.26iT - 19T^{2} \) |
| 23 | \( 1 + (6.37 + 3.68i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-8.58 + 4.95i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.76 + 2.17i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.23T + 37T^{2} \) |
| 41 | \( 1 + (-0.489 + 0.848i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0468 + 0.0811i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.86 + 3.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 5.61iT - 53T^{2} \) |
| 59 | \( 1 + (-0.620 + 1.07i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.45 - 4.30i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.21 - 7.30i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.9iT - 71T^{2} \) |
| 73 | \( 1 + 15.4iT - 73T^{2} \) |
| 79 | \( 1 + (-1.21 - 2.10i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.16 - 5.47i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + (-5.02 + 2.90i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.466100127866270770117170051601, −7.900073835477929036739921330784, −7.42288114536348377118397174689, −6.17244963017645368851916201810, −5.71716520041152118007817091227, −4.72905550461278675253000263943, −3.98751981363001395921611795703, −3.04473712441587698653316311489, −1.84295108771635792312524174321, −0.857562841327060150667044903072,
1.11619022990265062072068415075, 2.09406838952321559196613060093, 3.08451543914563120467899245940, 4.33393511980034025889833200115, 4.77555523805337873142062385397, 5.66978168524047299791966584699, 6.66635697463566648852281692507, 7.27250310727860142550304555683, 8.025907811188345315170466789613, 8.772345916747306683764584809274