L(s) = 1 | + 1.79·3-s + 0.791i·5-s + 2·7-s + 0.208·9-s − 0.791·11-s − 3.79i·13-s + 1.41i·15-s − 7.58i·17-s − 1.58i·19-s + 3.58·21-s − 3.79i·23-s + 4.37·25-s − 5.00·27-s + 3.79i·29-s − 8.37i·31-s + ⋯ |
L(s) = 1 | + 1.03·3-s + 0.353i·5-s + 0.755·7-s + 0.0695·9-s − 0.238·11-s − 1.05i·13-s + 0.365i·15-s − 1.83i·17-s − 0.363i·19-s + 0.781·21-s − 0.790i·23-s + 0.874·25-s − 0.962·27-s + 0.704i·29-s − 1.50i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.575367079\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.575367079\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + (4 + 4.58i)T \) |
good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 5 | \( 1 - 0.791iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 0.791T + 11T^{2} \) |
| 13 | \( 1 + 3.79iT - 13T^{2} \) |
| 17 | \( 1 + 7.58iT - 17T^{2} \) |
| 19 | \( 1 + 1.58iT - 19T^{2} \) |
| 23 | \( 1 + 3.79iT - 23T^{2} \) |
| 29 | \( 1 - 3.79iT - 29T^{2} \) |
| 31 | \( 1 + 8.37iT - 31T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 + 1.58iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 - 6.37T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 4.37T + 73T^{2} \) |
| 79 | \( 1 - 8.20iT - 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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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
−8.963633202898684824016295278464, −7.941962040623031580448230890107, −7.65507071854988929620190849401, −6.76368052075671174572480482190, −5.58485114066878824582000219282, −4.92005655641671058181415552312, −3.87434693572374988689400921228, −2.69546947526028342329291778800, −2.53663243174114953014976396468, −0.77704852772121285920396392283,
1.47573615914282718491903946366, 2.17971091804447593670885570866, 3.40074669212412105033369102525, 4.11310620312164614051509644756, 5.03204682960748304176726797506, 5.94018237902544048572204496104, 6.89625609451918823023688553547, 7.85544872709689132232965460256, 8.361606365622624874771662992456, 8.886662295360963310553965467623