| L(s) = 1 | + 2·3-s + 5-s + 3·9-s − 4·11-s − 4·13-s + 2·15-s + 2·19-s + 4·23-s + 10·27-s + 20·29-s + 4·31-s − 8·33-s + 2·37-s − 8·39-s + 24·41-s − 8·43-s + 3·45-s + 4·47-s − 2·53-s − 4·55-s + 4·57-s + 10·59-s + 6·61-s − 4·65-s − 4·67-s + 8·69-s − 24·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 9-s − 1.20·11-s − 1.10·13-s + 0.516·15-s + 0.458·19-s + 0.834·23-s + 1.92·27-s + 3.71·29-s + 0.718·31-s − 1.39·33-s + 0.328·37-s − 1.28·39-s + 3.74·41-s − 1.21·43-s + 0.447·45-s + 0.583·47-s − 0.274·53-s − 0.539·55-s + 0.529·57-s + 1.30·59-s + 0.768·61-s − 0.496·65-s − 0.488·67-s + 0.963·69-s − 2.84·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.604934996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.604934996\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.166840609803836950702115783097, −8.960421425907274738899532032580, −8.717468482294938447184205808120, −8.148329581601486044649937816036, −7.83692077214152721369993167709, −7.60521622899650995697235994790, −7.03850435890897415392504555493, −6.78058600754756636863006795444, −6.24822022538993410976362076829, −5.84306199818601759953795178332, −5.17804451240083742098851362886, −4.89808377454540728223694934966, −4.35748300733408916446237408824, −4.30198677351558779407602058559, −3.18054425471863492080787029405, −2.87229291914643712945775868962, −2.59456041597649457527616450044, −2.35848539388127365043949055436, −1.23928681899750154712394062183, −0.806958019465052529666861126641,
0.806958019465052529666861126641, 1.23928681899750154712394062183, 2.35848539388127365043949055436, 2.59456041597649457527616450044, 2.87229291914643712945775868962, 3.18054425471863492080787029405, 4.30198677351558779407602058559, 4.35748300733408916446237408824, 4.89808377454540728223694934966, 5.17804451240083742098851362886, 5.84306199818601759953795178332, 6.24822022538993410976362076829, 6.78058600754756636863006795444, 7.03850435890897415392504555493, 7.60521622899650995697235994790, 7.83692077214152721369993167709, 8.148329581601486044649937816036, 8.717468482294938447184205808120, 8.960421425907274738899532032580, 9.166840609803836950702115783097
