| L(s) = 1 | + 3-s − 5·5-s + 27·9-s + 7·11-s + 46·13-s − 5·15-s − 25·17-s − 62·19-s + 86·23-s + 80·27-s − 58·29-s − 12·31-s + 7·33-s + 150·37-s + 46·39-s − 408·41-s − 356·43-s − 135·45-s + 33·47-s − 25·51-s − 452·53-s − 35·55-s − 62·57-s + 120·59-s + 920·61-s − 230·65-s + 300·67-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 0.447·5-s + 9-s + 0.191·11-s + 0.981·13-s − 0.0860·15-s − 0.356·17-s − 0.748·19-s + 0.779·23-s + 0.570·27-s − 0.371·29-s − 0.0695·31-s + 0.0369·33-s + 0.666·37-s + 0.188·39-s − 1.55·41-s − 1.26·43-s − 0.447·45-s + 0.102·47-s − 0.0686·51-s − 1.17·53-s − 0.0858·55-s − 0.144·57-s + 0.264·59-s + 1.93·61-s − 0.438·65-s + 0.547·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.558202646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.558202646\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 26 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 7 T - 1282 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 23 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 25 T - 4288 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 62 T - 3015 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 86 T - 4771 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 12 T - 29647 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 150 T - 28153 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 204 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 178 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 33 T - 102734 T^{2} - 33 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 452 T + 55427 T^{2} + 452 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 120 T - 190979 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 920 T + 619419 T^{2} - 920 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 300 T - 210763 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 520 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 370 T - 252117 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1013 T + 533130 T^{2} - 1013 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 636 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 292 T - 619705 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1381 T + p^{3} T^{2} )^{2} \) |
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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
−9.680829146742512283245697314442, −9.557362254803801152961214019553, −8.776158133476374440988373415558, −8.720610361147025841921867721876, −8.109516767940463180395512845363, −7.87517708968028144757327102986, −7.26682636281730129561541847335, −6.82098372492757166277130631601, −6.38788337774839780455465706008, −6.32519153791153022071215226233, −5.27421459406670266095904731332, −5.07058406709113345321093074028, −4.48882180682028855818488605379, −4.01007950189869516411631181433, −3.43499261969073760872123636792, −3.30222370538920811756429902841, −2.13292438950492262601301988211, −1.95076871011351465883411178091, −1.03431446025681599126858838319, −0.54768538601242965769033525709,
0.54768538601242965769033525709, 1.03431446025681599126858838319, 1.95076871011351465883411178091, 2.13292438950492262601301988211, 3.30222370538920811756429902841, 3.43499261969073760872123636792, 4.01007950189869516411631181433, 4.48882180682028855818488605379, 5.07058406709113345321093074028, 5.27421459406670266095904731332, 6.32519153791153022071215226233, 6.38788337774839780455465706008, 6.82098372492757166277130631601, 7.26682636281730129561541847335, 7.87517708968028144757327102986, 8.109516767940463180395512845363, 8.720610361147025841921867721876, 8.776158133476374440988373415558, 9.557362254803801152961214019553, 9.680829146742512283245697314442
