| L(s) = 1 | − 13·4-s + 105·16-s + 234·17-s − 156·23-s − 247·25-s + 282·29-s − 208·43-s − 494·49-s − 186·53-s + 290·61-s − 533·64-s − 3.04e3·68-s + 2.55e3·79-s + 2.02e3·92-s + 3.21e3·100-s + 858·101-s − 364·103-s + 3.01e3·107-s + 1.37e3·113-s − 3.66e3·116-s − 2.47e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.62·4-s + 1.64·16-s + 3.33·17-s − 1.41·23-s − 1.97·25-s + 1.80·29-s − 0.737·43-s − 1.44·49-s − 0.482·53-s + 0.608·61-s − 1.04·64-s − 5.42·68-s + 3.63·79-s + 2.29·92-s + 3.21·100-s + 0.845·101-s − 0.348·103-s + 2.72·107-s + 1.14·113-s − 2.93·116-s − 1.85·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.365073391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.365073391\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 13 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 247 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 494 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2470 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 117 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 650 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 78 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 141 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 35282 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 80639 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 63895 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 104 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 116818 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 93 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 330070 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 145 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 16822 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 400478 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 567359 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1276 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 519766 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 455650 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1784978 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.327486932291249285724095862797, −9.040852406691120715305276075180, −8.206063465790956996325082526317, −8.189244979245667943466975715228, −7.77423134477657699827200500138, −7.75316063070374836273976246608, −6.83742857468351378204704595839, −6.35093014288600729141919116372, −5.77759699241434142784358207487, −5.70450125610762439733533555459, −5.02172039165984458568670852084, −4.87402652743255754463629027390, −4.16889766828211972020294764055, −3.82991968795437775432555786804, −3.22096572160771695183392359924, −3.21004743394228165398118068961, −2.03790170726959255520540796148, −1.57727971356795834644781448684, −0.68555257622848918656260438554, −0.57034227746350611523180720085,
0.57034227746350611523180720085, 0.68555257622848918656260438554, 1.57727971356795834644781448684, 2.03790170726959255520540796148, 3.21004743394228165398118068961, 3.22096572160771695183392359924, 3.82991968795437775432555786804, 4.16889766828211972020294764055, 4.87402652743255754463629027390, 5.02172039165984458568670852084, 5.70450125610762439733533555459, 5.77759699241434142784358207487, 6.35093014288600729141919116372, 6.83742857468351378204704595839, 7.75316063070374836273976246608, 7.77423134477657699827200500138, 8.189244979245667943466975715228, 8.206063465790956996325082526317, 9.040852406691120715305276075180, 9.327486932291249285724095862797
