L(s) = 1 | + 2.23i·5-s + 6.51i·7-s − 16.5i·11-s − 12.7·13-s − 15.5i·17-s − 21.3i·19-s + (−21.7 + 7.49i)23-s − 5.00·25-s − 21.6·29-s + 61.2·31-s − 14.5·35-s + 29.9i·37-s − 57.4·41-s − 61.2i·43-s + 72.3·47-s + ⋯ |
L(s) = 1 | + 0.447i·5-s + 0.930i·7-s − 1.50i·11-s − 0.977·13-s − 0.911i·17-s − 1.12i·19-s + (−0.945 + 0.325i)23-s − 0.200·25-s − 0.745·29-s + 1.97·31-s − 0.416·35-s + 0.809i·37-s − 1.40·41-s − 1.42i·43-s + 1.54·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.003723698\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.003723698\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (21.7 - 7.49i)T \) |
good | 7 | \( 1 - 6.51iT - 49T^{2} \) |
| 11 | \( 1 + 16.5iT - 121T^{2} \) |
| 13 | \( 1 + 12.7T + 169T^{2} \) |
| 17 | \( 1 + 15.5iT - 289T^{2} \) |
| 19 | \( 1 + 21.3iT - 361T^{2} \) |
| 29 | \( 1 + 21.6T + 841T^{2} \) |
| 31 | \( 1 - 61.2T + 961T^{2} \) |
| 37 | \( 1 - 29.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 57.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 61.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 72.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 98.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 40.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 48.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 70.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 62.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 13.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 105. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 16.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 148. iT - 9.40e3T^{2} \) |
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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
−8.677732871439320815897197372233, −7.69284341147556827117824776322, −7.03444023348326182614261926643, −6.18964594282846484930433337887, −5.56330285566272437549275305841, −4.86886685657986468902543018347, −3.80474159342200030134032392061, −2.72423505672091047010103991429, −2.49686974438335328044096925341, −0.880109551025276287487883269110,
0.24130188812397303617363024014, 1.55399664588242527181668861114, 2.23081962025758032563416421800, 3.58886863116810590320646834217, 4.33582552145984600694918231524, 4.80237178205701071267164897656, 5.83847174279265297240911758278, 6.63859244257836940226129254249, 7.41727620413942956033151360114, 7.906079999878141712115260555416