| L(s) = 1 | − 2-s − 5-s − 2·7-s + 8-s + 10-s − 11-s + 2·13-s + 2·14-s − 16-s + 19-s + 22-s + 23-s − 2·26-s + 2·35-s − 37-s − 38-s − 40-s + 2·41-s − 46-s + 47-s + 3·49-s + 53-s + 55-s − 2·56-s + 2·59-s + 64-s − 2·65-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s − 2·7-s + 8-s + 10-s − 11-s + 2·13-s + 2·14-s − 16-s + 19-s + 22-s + 23-s − 2·26-s + 2·35-s − 37-s − 38-s − 40-s + 2·41-s − 46-s + 47-s + 3·49-s + 53-s + 55-s − 2·56-s + 2·59-s + 64-s − 2·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4616584813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4616584813\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.272230059152160766884585327618, −8.844047786716949057066935965556, −8.537304255024880709022964785813, −8.381583803784434653284213351961, −7.71365080051732131397662895713, −7.46268586290582438794133422486, −7.01812159067689128591763698961, −6.95583966826807034436603210237, −6.19738271677252500345239190107, −5.78471624221517926152583996373, −5.66026845929300098782881070959, −4.93174851592307173909917297673, −4.44901820758347014391162706101, −3.87246326021413010558483355856, −3.47077487476381426223089574212, −3.44213604577516428566122197221, −2.64355183091304648818436232957, −2.15443909938462274359583727140, −0.980925029934685749612827896588, −0.71763195939815427300915388734,
0.71763195939815427300915388734, 0.980925029934685749612827896588, 2.15443909938462274359583727140, 2.64355183091304648818436232957, 3.44213604577516428566122197221, 3.47077487476381426223089574212, 3.87246326021413010558483355856, 4.44901820758347014391162706101, 4.93174851592307173909917297673, 5.66026845929300098782881070959, 5.78471624221517926152583996373, 6.19738271677252500345239190107, 6.95583966826807034436603210237, 7.01812159067689128591763698961, 7.46268586290582438794133422486, 7.71365080051732131397662895713, 8.381583803784434653284213351961, 8.537304255024880709022964785813, 8.844047786716949057066935965556, 9.272230059152160766884585327618
