| L(s) = 1 | + 2-s − 3-s − 4·5-s − 6-s + 7-s − 8-s − 4·10-s + 4·11-s − 7·13-s + 14-s + 4·15-s − 16-s − 7·17-s − 2·19-s − 21-s + 4·22-s + 23-s + 24-s + 2·25-s − 7·26-s + 27-s + 2·29-s + 4·30-s − 18·31-s − 4·33-s − 7·34-s − 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1.78·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 1.26·10-s + 1.20·11-s − 1.94·13-s + 0.267·14-s + 1.03·15-s − 1/4·16-s − 1.69·17-s − 0.458·19-s − 0.218·21-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 2/5·25-s − 1.37·26-s + 0.192·27-s + 0.371·29-s + 0.730·30-s − 3.23·31-s − 0.696·33-s − 1.20·34-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4869817816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4869817816\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−11.34158752982643437939785428671, −10.80529507801255765824642357014, −10.41134461087545982118472918663, −9.511576942030541412130134295292, −9.199742222134130412445483437910, −8.978480425123472703572660406686, −8.218561257307482672033682800642, −7.81221791838822531039387653706, −7.33803885625386128591416180320, −6.88700727658138136191215866695, −6.69770790918759634308999076011, −5.71462156459475568842742310562, −5.47648572000703580374583960222, −4.62553111551699653649198192288, −4.48732756969274897181065046809, −3.85197468361206663828275367992, −3.67833373829573721776399603891, −2.55804641093132939198978389761, −1.95500335569753430586389179570, −0.36138729858545339694152960983,
0.36138729858545339694152960983, 1.95500335569753430586389179570, 2.55804641093132939198978389761, 3.67833373829573721776399603891, 3.85197468361206663828275367992, 4.48732756969274897181065046809, 4.62553111551699653649198192288, 5.47648572000703580374583960222, 5.71462156459475568842742310562, 6.69770790918759634308999076011, 6.88700727658138136191215866695, 7.33803885625386128591416180320, 7.81221791838822531039387653706, 8.218561257307482672033682800642, 8.978480425123472703572660406686, 9.199742222134130412445483437910, 9.511576942030541412130134295292, 10.41134461087545982118472918663, 10.80529507801255765824642357014, 11.34158752982643437939785428671
