L(s) = 1 | − 4·5-s + 10·7-s + 10·11-s − 24.2i·13-s − 10·17-s − 19·19-s − 20·23-s − 9·25-s + 34.6i·29-s − 17.3i·31-s − 40·35-s − 10.3i·37-s − 34.6i·41-s + 10·43-s − 80·47-s + ⋯ |
L(s) = 1 | − 0.800·5-s + 1.42·7-s + 0.909·11-s − 1.86i·13-s − 0.588·17-s − 19-s − 0.869·23-s − 0.359·25-s + 1.19i·29-s − 0.558i·31-s − 1.14·35-s − 0.280i·37-s − 0.844i·41-s + 0.232·43-s − 1.70·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3731935293\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3731935293\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 19T \) |
good | 5 | \( 1 + 4T + 25T^{2} \) |
| 7 | \( 1 - 10T + 49T^{2} \) |
| 11 | \( 1 - 10T + 121T^{2} \) |
| 13 | \( 1 + 24.2iT - 169T^{2} \) |
| 17 | \( 1 + 10T + 289T^{2} \) |
| 23 | \( 1 + 20T + 529T^{2} \) |
| 29 | \( 1 - 34.6iT - 841T^{2} \) |
| 31 | \( 1 + 17.3iT - 961T^{2} \) |
| 37 | \( 1 + 10.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 34.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10T + 1.84e3T^{2} \) |
| 47 | \( 1 + 80T + 2.20e3T^{2} \) |
| 53 | \( 1 + 41.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 34.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 10T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 17.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 70T + 6.88e3T^{2} \) |
| 89 | \( 1 - 103. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 76.2iT - 9.40e3T^{2} \) |
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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
−8.275894058089597159868054884042, −7.70419885108029885613329107263, −6.88093828368026515409070534189, −5.85803290473776230659147779265, −5.09827804762983624473892201321, −4.23558673262339735914459964559, −3.61096462915241623382695676982, −2.34942641025673812072461627161, −1.29889896153095985114723183451, −0.085063885152692211467309290560,
1.54385997192990941027552130606, 2.11347530713007540101659571774, 3.71485947343092483477021546398, 4.42214055269399308662618963456, 4.71596957154463816108555574343, 6.21384291701002508986663312584, 6.65767758222565868448497478835, 7.71532754222860423244608047131, 8.182192272491092724713868251433, 8.939740459607827378890608954137