| L(s) = 1 | + 2·2-s − 3-s − 4-s − 2·6-s + 2·7-s − 8·8-s − 4·9-s + 5·11-s + 12-s − 3·13-s + 4·14-s − 7·16-s − 17-s − 8·18-s − 19-s − 2·21-s + 10·22-s + 8·23-s + 8·24-s − 6·26-s + 6·27-s − 2·28-s − 3·29-s − 12·31-s + 14·32-s − 5·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s − 1/2·4-s − 0.816·6-s + 0.755·7-s − 2.82·8-s − 4/3·9-s + 1.50·11-s + 0.288·12-s − 0.832·13-s + 1.06·14-s − 7/4·16-s − 0.242·17-s − 1.88·18-s − 0.229·19-s − 0.436·21-s + 2.13·22-s + 1.66·23-s + 1.63·24-s − 1.17·26-s + 1.15·27-s − 0.377·28-s − 0.557·29-s − 2.15·31-s + 2.47·32-s − 0.870·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 241 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 37 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 53 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 103 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 113 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T - 17 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T - 3 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 135 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_4$ | \( 1 + 2 T - 86 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 29 T + 375 T^{2} + 29 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 302 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 30 T + 414 T^{2} + 30 p T^{3} + p^{2} T^{4} \) |
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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
−7.88198068261627625992172842752, −7.49695306140242165633505722356, −7.05578062424892189399876822957, −6.64359412473683739586490222084, −6.25653433758397850276363006338, −5.99616195717741718684813062246, −5.47300565336809219123978818398, −5.44257956310430794762720395613, −4.92999759025866542459294126814, −4.83001842352504616109889459924, −4.15440927684309254352472626099, −4.12398218071431491769463292515, −3.46056638461975427499994014528, −3.33319661271851121499881369765, −2.56158377187030169526910789950, −2.52235164664748978205804986170, −1.48337751266198804670223163565, −1.06705379941081573497292323040, 0, 0,
1.06705379941081573497292323040, 1.48337751266198804670223163565, 2.52235164664748978205804986170, 2.56158377187030169526910789950, 3.33319661271851121499881369765, 3.46056638461975427499994014528, 4.12398218071431491769463292515, 4.15440927684309254352472626099, 4.83001842352504616109889459924, 4.92999759025866542459294126814, 5.44257956310430794762720395613, 5.47300565336809219123978818398, 5.99616195717741718684813062246, 6.25653433758397850276363006338, 6.64359412473683739586490222084, 7.05578062424892189399876822957, 7.49695306140242165633505722356, 7.88198068261627625992172842752
