| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s − 9-s + 10-s − 13-s + 16-s + 9·17-s + 18-s − 20-s − 4·25-s + 26-s + 2·29-s − 32-s − 9·34-s − 36-s + 40-s − 8·41-s + 45-s − 2·49-s + 4·50-s − 52-s − 12·53-s − 2·58-s − 6·61-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 1/3·9-s + 0.316·10-s − 0.277·13-s + 1/4·16-s + 2.18·17-s + 0.235·18-s − 0.223·20-s − 4/5·25-s + 0.196·26-s + 0.371·29-s − 0.176·32-s − 1.54·34-s − 1/6·36-s + 0.158·40-s − 1.24·41-s + 0.149·45-s − 2/7·49-s + 0.565·50-s − 0.138·52-s − 1.64·53-s − 0.262·58-s − 0.768·61-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.040945589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040945589\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 39 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - T + p T^{2} ) \) |
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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
−7.957280666540630038117887845150, −7.60477782838362755023963098759, −7.41536149107185106362253774986, −6.83510049285865674356133816387, −6.20147080174210806494925265124, −5.94311844478485826871141753456, −5.46834594555644837607768960869, −4.84533270837997302217977183730, −4.52071818674028892543550773451, −3.55049261748288515610495692298, −3.38969426365527052871400305058, −2.88474985670269204800389596300, −2.00961077241906285225017557549, −1.43411828470338557013146422851, −0.52853842907278111646901727161,
0.52853842907278111646901727161, 1.43411828470338557013146422851, 2.00961077241906285225017557549, 2.88474985670269204800389596300, 3.38969426365527052871400305058, 3.55049261748288515610495692298, 4.52071818674028892543550773451, 4.84533270837997302217977183730, 5.46834594555644837607768960869, 5.94311844478485826871141753456, 6.20147080174210806494925265124, 6.83510049285865674356133816387, 7.41536149107185106362253774986, 7.60477782838362755023963098759, 7.957280666540630038117887845150
