| L(s) = 1 | − 2-s − 3-s − 2·5-s + 6-s + 8-s + 2·10-s + 4·11-s − 12·13-s + 2·15-s − 16-s + 2·17-s − 4·19-s − 4·22-s − 8·23-s − 24-s + 5·25-s + 12·26-s + 27-s − 4·29-s − 2·30-s − 4·33-s − 2·34-s + 10·37-s + 4·38-s + 12·39-s − 2·40-s + 12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.894·5-s + 0.408·6-s + 0.353·8-s + 0.632·10-s + 1.20·11-s − 3.32·13-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.852·22-s − 1.66·23-s − 0.204·24-s + 25-s + 2.35·26-s + 0.192·27-s − 0.742·29-s − 0.365·30-s − 0.696·33-s − 0.342·34-s + 1.64·37-s + 0.648·38-s + 1.92·39-s − 0.316·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3552696109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3552696109\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−12.13088943882806020005347097349, −11.52139013956716966692339485737, −11.21874463868288977456539625125, −10.48893098039013274090130252498, −10.08241260517181695135022430656, −9.571322768419506760571051977178, −9.408518829385937771906293456981, −8.766647804035256629565796305818, −7.88157836693051664150520105718, −7.80714630839587868580542486676, −7.40517947442939017957074667149, −6.49004749384355708572760411240, −6.47275686945982252098654723090, −5.32582021709298821923335436848, −4.97636167220636002120561114835, −4.21536856473896139180963478796, −3.93988941373864831048355907815, −2.71582742669109014304947578532, −2.04072986579305853023600278227, −0.50536055101104460612865375100,
0.50536055101104460612865375100, 2.04072986579305853023600278227, 2.71582742669109014304947578532, 3.93988941373864831048355907815, 4.21536856473896139180963478796, 4.97636167220636002120561114835, 5.32582021709298821923335436848, 6.47275686945982252098654723090, 6.49004749384355708572760411240, 7.40517947442939017957074667149, 7.80714630839587868580542486676, 7.88157836693051664150520105718, 8.766647804035256629565796305818, 9.408518829385937771906293456981, 9.571322768419506760571051977178, 10.08241260517181695135022430656, 10.48893098039013274090130252498, 11.21874463868288977456539625125, 11.52139013956716966692339485737, 12.13088943882806020005347097349
