L(s) = 1 | + (−1 + i)5-s + 4i·7-s + (−4 + 4i)11-s + (3 + 3i)13-s − 6·17-s + (4 + 4i)19-s + 8i·23-s + 3i·25-s + (−3 − 3i)29-s − 4·31-s + (−4 − 4i)35-s + (1 − i)37-s + 2i·41-s + (4 − 4i)43-s + 8·47-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.447i)5-s + 1.51i·7-s + (−1.20 + 1.20i)11-s + (0.832 + 0.832i)13-s − 1.45·17-s + (0.917 + 0.917i)19-s + 1.66i·23-s + 0.600i·25-s + (−0.557 − 0.557i)29-s − 0.718·31-s + (−0.676 − 0.676i)35-s + (0.164 − 0.164i)37-s + 0.312i·41-s + (0.609 − 0.609i)43-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.110692162\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.110692162\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1 - i)T - 5iT^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + (4 - 4i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (-4 - 4i)T + 19iT^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + (3 + 3i)T + 29iT^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-1 + i)T - 37iT^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 + (-4 + 4i)T - 43iT^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + (-7 + 7i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (3 + 3i)T + 61iT^{2} \) |
| 67 | \( 1 + (8 + 8i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + (-4 - 4i)T + 83iT^{2} \) |
| 89 | \( 1 - 16iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.870341549955615473674684896071, −7.82737365338261538371708008248, −7.43088585351909865083665158145, −6.58252006786973303046804234986, −5.63468434581858936814025940839, −5.26806914846655241723377763075, −4.16376595904785657381606227810, −3.39975543309232333072261760163, −2.31736008334435034936321024512, −1.82125963988396801340849802761,
0.38952170625411894807859087068, 0.852836790050615844638297801424, 2.52948468646778236519954234190, 3.33940838626336626233591849495, 4.20955607052196833668186563818, 4.78012559087226148824548772156, 5.71608040639416692644027432144, 6.48448729505008980851536982593, 7.39712626601731869178495080939, 7.80286226383048528224690664544