| L(s) = 1 | − 2·2-s − 5·3-s − 5·5-s + 10·6-s + 8·8-s + 27·9-s + 10·10-s + 11-s + 14·13-s + 25·15-s − 16·16-s + 51·17-s − 54·18-s − 30·19-s − 2·22-s + 50·23-s − 40·24-s − 28·26-s − 280·27-s + 158·29-s − 50·30-s + 212·31-s − 5·33-s − 102·34-s + 190·37-s + 60·38-s − 70·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.962·3-s − 0.447·5-s + 0.680·6-s + 0.353·8-s + 9-s + 0.316·10-s + 0.0274·11-s + 0.298·13-s + 0.430·15-s − 1/4·16-s + 0.727·17-s − 0.707·18-s − 0.362·19-s − 0.0193·22-s + 0.453·23-s − 0.340·24-s − 0.211·26-s − 1.99·27-s + 1.01·29-s − 0.304·30-s + 1.22·31-s − 0.0263·33-s − 0.514·34-s + 0.844·37-s + 0.256·38-s − 0.287·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9403416614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9403416614\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 5 T - 2 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 1330 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 p T - 8 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T - 5959 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 50 T - 9667 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 79 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 212 T + 15153 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 190 T - 14553 T^{2} - 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 308 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 422 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 121 T - 89182 T^{2} + 121 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 664 T + 292019 T^{2} + 664 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 628 T + 189005 T^{2} + 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 684 T + 240875 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 1056 T + 814373 T^{2} + 1056 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 744 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 726 T + 138059 T^{2} + 726 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 407 T - 327390 T^{2} - 407 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 644 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 880 T + 69431 T^{2} - 880 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1351 T + p^{3} T^{2} )^{2} \) |
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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
−10.68568695518191094307827888966, −10.40735718174245443722740300830, −9.906871897203828025911996272176, −9.420265419470492253154915587356, −9.211242354756973159055869950559, −8.305432142388126102033190474989, −8.204765299443246914079084568373, −7.58613451247077272727567947871, −7.25313859705070681781003111637, −6.59887589426780255821289044212, −6.16302055713008480509346158311, −5.80024357774668543848818205459, −4.88908399384018191117639650450, −4.77536450229043684896817770872, −4.03313542334872964373845386122, −3.49605093602361083736444209012, −2.69025205594616876780884454139, −1.75011793743177162466497783968, −1.03518123319900977051991247478, −0.45704922648981625018018499891,
0.45704922648981625018018499891, 1.03518123319900977051991247478, 1.75011793743177162466497783968, 2.69025205594616876780884454139, 3.49605093602361083736444209012, 4.03313542334872964373845386122, 4.77536450229043684896817770872, 4.88908399384018191117639650450, 5.80024357774668543848818205459, 6.16302055713008480509346158311, 6.59887589426780255821289044212, 7.25313859705070681781003111637, 7.58613451247077272727567947871, 8.204765299443246914079084568373, 8.305432142388126102033190474989, 9.211242354756973159055869950559, 9.420265419470492253154915587356, 9.906871897203828025911996272176, 10.40735718174245443722740300830, 10.68568695518191094307827888966
