| L(s) = 1 | − 2·2-s + 5·3-s + 12·5-s − 10·6-s + 16·7-s + 8·8-s + 27·9-s − 24·10-s + 18·11-s − 26·13-s − 32·14-s + 60·15-s − 16·16-s − 114·17-s − 54·18-s − 133·19-s + 80·21-s − 36·22-s + 78·23-s + 40·24-s + 125·25-s + 52·26-s + 280·27-s + 204·29-s − 120·30-s + 196·31-s + 90·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.962·3-s + 1.07·5-s − 0.680·6-s + 0.863·7-s + 0.353·8-s + 9-s − 0.758·10-s + 0.493·11-s − 0.554·13-s − 0.610·14-s + 1.03·15-s − 1/4·16-s − 1.62·17-s − 0.707·18-s − 1.60·19-s + 0.831·21-s − 0.348·22-s + 0.707·23-s + 0.340·24-s + 25-s + 0.392·26-s + 1.99·27-s + 1.30·29-s − 0.730·30-s + 1.13·31-s + 0.474·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.735403532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.735403532\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 19 | $C_2$ | \( 1 + 7 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T - 2 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 12 T + 19 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 9 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 p T + p^{3} T^{2} )( 1 + 7 p T + p^{3} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 114 T + 8083 T^{2} + 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 204 T + 17227 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 98 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 334 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 177 T - 37592 T^{2} + 177 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 316 T + 20349 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 492 T + 138241 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 678 T + 310807 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 579 T + 129862 T^{2} - 579 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 352 T - 103077 T^{2} - 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 755 T + 269262 T^{2} + 755 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 357875 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 145 T - 367992 T^{2} - 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 p T + p^{3} T^{2} )( 1 + 13 p T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 567 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 114 T - 691973 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 943 T - 23424 T^{2} - 943 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.79344673072152904997313759790, −15.76774923889544842069056797012, −14.60973975809949612176928169533, −14.52408829033646498089762438057, −13.79477012140174774009642549862, −13.34512977626873638597211088267, −12.61974175118189989909971753823, −12.00217775923006031960173849819, −10.71420165249823358998415755773, −10.62611418324146687729592670656, −9.814875581255367554197956981493, −8.954429043077708115769500661252, −8.706774222914865863429419266714, −8.145497968852400726971723120636, −6.79763651794006797375943152556, −6.67426778657871342886957032830, −4.97008215825124048539788859130, −4.38639626357381357373537488245, −2.61049759785726038636040849688, −1.62139973029177866557976345889,
1.62139973029177866557976345889, 2.61049759785726038636040849688, 4.38639626357381357373537488245, 4.97008215825124048539788859130, 6.67426778657871342886957032830, 6.79763651794006797375943152556, 8.145497968852400726971723120636, 8.706774222914865863429419266714, 8.954429043077708115769500661252, 9.814875581255367554197956981493, 10.62611418324146687729592670656, 10.71420165249823358998415755773, 12.00217775923006031960173849819, 12.61974175118189989909971753823, 13.34512977626873638597211088267, 13.79477012140174774009642549862, 14.52408829033646498089762438057, 14.60973975809949612176928169533, 15.76774923889544842069056797012, 15.79344673072152904997313759790
