| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 2·17-s + 2·19-s − 2·23-s − 6·32-s − 4·34-s − 4·38-s + 4·46-s + 2·47-s + 2·61-s + 7·64-s + 6·68-s + 6·76-s + 4·83-s − 6·92-s − 4·94-s + 4·107-s − 2·109-s − 2·113-s − 4·122-s + 127-s − 8·128-s + 131-s − 8·136-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 2·17-s + 2·19-s − 2·23-s − 6·32-s − 4·34-s − 4·38-s + 4·46-s + 2·47-s + 2·61-s + 7·64-s + 6·68-s + 6·76-s + 4·83-s − 6·92-s − 4·94-s + 4·107-s − 2·109-s − 2·113-s − 4·122-s + 127-s − 8·128-s + 131-s − 8·136-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7179313801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7179313801\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 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
−8.763907658950581860575461707022, −8.747506268940613382712887968863, −7.949276505332593696836538162918, −7.896773680884115952924095968678, −7.54005616866590754321907977609, −7.49948590138733128515487477591, −6.75788248891037997492784621781, −6.55877104087657494037607875114, −5.96427359175341310717797828612, −5.73867384980883470554502285022, −5.33240190214868689701780887518, −5.08251563771190697598871274012, −3.90029188327220822394473588323, −3.85240628417037585804803315095, −3.12711639803258165308250889241, −2.99415304506268720952436637574, −2.05766774731570020841204835253, −2.05762052439398049788431617748, −1.00039767115762042892346774083, −0.901534009153222065044194060008,
0.901534009153222065044194060008, 1.00039767115762042892346774083, 2.05762052439398049788431617748, 2.05766774731570020841204835253, 2.99415304506268720952436637574, 3.12711639803258165308250889241, 3.85240628417037585804803315095, 3.90029188327220822394473588323, 5.08251563771190697598871274012, 5.33240190214868689701780887518, 5.73867384980883470554502285022, 5.96427359175341310717797828612, 6.55877104087657494037607875114, 6.75788248891037997492784621781, 7.49948590138733128515487477591, 7.54005616866590754321907977609, 7.896773680884115952924095968678, 7.949276505332593696836538162918, 8.747506268940613382712887968863, 8.763907658950581860575461707022
