| L(s) = 1 | − 3-s + 3·4-s + 9-s − 3·12-s + 2·13-s + 5·16-s + 6·25-s − 27-s + 3·36-s − 2·39-s − 8·43-s − 5·48-s + 49-s + 6·52-s + 4·61-s + 3·64-s − 6·75-s + 32·79-s + 81-s + 18·100-s + 16·103-s − 3·108-s + 2·117-s + 6·121-s + 127-s + 8·129-s + 131-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 3/2·4-s + 1/3·9-s − 0.866·12-s + 0.554·13-s + 5/4·16-s + 6/5·25-s − 0.192·27-s + 1/2·36-s − 0.320·39-s − 1.21·43-s − 0.721·48-s + 1/7·49-s + 0.832·52-s + 0.512·61-s + 3/8·64-s − 0.692·75-s + 3.60·79-s + 1/9·81-s + 9/5·100-s + 1.57·103-s − 0.288·108-s + 0.184·117-s + 6/11·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 223587 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223587 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.208684594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208684594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.037974216704180159543126631909, −8.356175078750261586430624145966, −8.132252088522676141887559266017, −7.27274257516820834559425411006, −7.17760217788793919270753307155, −6.52472878301895647420990933872, −6.26011193460154366739782569927, −5.78110689368608507428910017033, −5.05101314154979794048381255754, −4.71888444449924838616842819054, −3.71600133819353144201904115814, −3.31075497543291353482285565928, −2.50227594714982301545782982287, −1.88185952070879662660487292677, −0.995956301106271632064071395001,
0.995956301106271632064071395001, 1.88185952070879662660487292677, 2.50227594714982301545782982287, 3.31075497543291353482285565928, 3.71600133819353144201904115814, 4.71888444449924838616842819054, 5.05101314154979794048381255754, 5.78110689368608507428910017033, 6.26011193460154366739782569927, 6.52472878301895647420990933872, 7.17760217788793919270753307155, 7.27274257516820834559425411006, 8.132252088522676141887559266017, 8.356175078750261586430624145966, 9.037974216704180159543126631909
