L(s) = 1 | − 4-s + 16-s − 2·17-s − 2·19-s − 2·23-s − 2·47-s − 4·53-s + 2·61-s − 64-s + 2·68-s + 2·76-s + 2·92-s + 2·109-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 4-s + 16-s − 2·17-s − 2·19-s − 2·23-s − 2·47-s − 4·53-s + 2·61-s − 64-s + 2·68-s + 2·76-s + 2·92-s + 2·109-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3231541701\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3231541701\) |
\(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 53 | $C_1$ | \( ( 1 + T )^{4} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T^{4} \) |
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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
−8.840329471585470289009577905146, −8.464483206693136692419725031041, −8.307954698974186515567994325747, −7.950447178398915132708878322573, −7.64546647484322424397325663200, −6.88925762661635178074584972853, −6.61661409666853980558975589704, −6.40424146961632750196400319984, −5.83733642954096286456462120135, −5.75072337935010036189406773612, −4.80637539491969613091654765738, −4.69870004726746144078376948285, −4.43682765824532894580321547985, −4.04885870075158876883996776906, −3.43220991952788035647079883045, −3.21988968857554125625413749981, −2.21963929158248161733974633278, −2.09847707073508280553719848146, −1.56324844890515915473163182591, −0.32011811031841551413749580584,
0.32011811031841551413749580584, 1.56324844890515915473163182591, 2.09847707073508280553719848146, 2.21963929158248161733974633278, 3.21988968857554125625413749981, 3.43220991952788035647079883045, 4.04885870075158876883996776906, 4.43682765824532894580321547985, 4.69870004726746144078376948285, 4.80637539491969613091654765738, 5.75072337935010036189406773612, 5.83733642954096286456462120135, 6.40424146961632750196400319984, 6.61661409666853980558975589704, 6.88925762661635178074584972853, 7.64546647484322424397325663200, 7.950447178398915132708878322573, 8.307954698974186515567994325747, 8.464483206693136692419725031041, 8.840329471585470289009577905146