| L(s) = 1 | + 2·2-s − 5·5-s + 4·7-s − 8·8-s − 10·10-s + 42·11-s − 20·13-s + 8·14-s − 16·16-s − 186·17-s + 118·19-s + 84·22-s + 9·23-s − 40·26-s + 120·29-s − 47·31-s − 372·34-s − 20·35-s − 524·37-s + 236·38-s + 40·40-s + 126·41-s + 178·43-s + 18·46-s + 144·47-s + 343·49-s − 1.48e3·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s + 0.215·7-s − 0.353·8-s − 0.316·10-s + 1.15·11-s − 0.426·13-s + 0.152·14-s − 1/4·16-s − 2.65·17-s + 1.42·19-s + 0.814·22-s + 0.0815·23-s − 0.301·26-s + 0.768·29-s − 0.272·31-s − 1.87·34-s − 0.0965·35-s − 2.32·37-s + 1.00·38-s + 0.158·40-s + 0.479·41-s + 0.631·43-s + 0.0576·46-s + 0.446·47-s + 49-s − 3.84·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8539027528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8539027528\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 4 T - 327 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 42 T + 433 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T - 1797 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 93 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 59 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9 T - 12086 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 120 T - 9989 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 47 T - 27582 T^{2} + 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 262 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 126 T - 53045 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 178 T - 47823 T^{2} - 178 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 144 T - 83087 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 741 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 444 T - 8243 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 221 T - 178140 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 538 T - 11319 T^{2} - 538 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 690 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1126 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 665 T - 50814 T^{2} + 665 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 75 T - 566162 T^{2} - 75 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1086 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1544 T + 1471263 T^{2} + 1544 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.35244089591395288850059947338, −9.310517363530664891794130067158, −9.305636941230849243106292939456, −8.716974368978870460528275916347, −8.703087283481579300676929845677, −7.71849758412349843201552210072, −7.51465468573401674226038709066, −6.96254969412827527707059194670, −6.57794196636827774390423265585, −6.17027099838268383320407887343, −5.65332578864454022220749837141, −4.93664643501540634929347995162, −4.65785461220159410841570971472, −4.26648962104952081694893654842, −3.79351773242808896595876118518, −3.07844079670064303598111443721, −2.75073584178580522709338437894, −1.78585113414510919051822459903, −1.37046784436751794002850280141, −0.21669737127033303428434861931,
0.21669737127033303428434861931, 1.37046784436751794002850280141, 1.78585113414510919051822459903, 2.75073584178580522709338437894, 3.07844079670064303598111443721, 3.79351773242808896595876118518, 4.26648962104952081694893654842, 4.65785461220159410841570971472, 4.93664643501540634929347995162, 5.65332578864454022220749837141, 6.17027099838268383320407887343, 6.57794196636827774390423265585, 6.96254969412827527707059194670, 7.51465468573401674226038709066, 7.71849758412349843201552210072, 8.703087283481579300676929845677, 8.716974368978870460528275916347, 9.305636941230849243106292939456, 9.310517363530664891794130067158, 10.35244089591395288850059947338
