| L(s) = 1 | + 2·4-s + 5-s + 4·7-s + 6·11-s + 4·13-s − 7·19-s + 2·20-s − 6·23-s + 8·28-s + 3·29-s + 10·31-s + 4·35-s + 16·37-s − 6·41-s + 4·43-s + 12·44-s + 6·47-s − 2·49-s + 8·52-s − 6·53-s + 6·55-s − 9·59-s + 7·61-s − 8·64-s + 4·65-s − 2·67-s − 9·71-s + ⋯ |
| L(s) = 1 | + 4-s + 0.447·5-s + 1.51·7-s + 1.80·11-s + 1.10·13-s − 1.60·19-s + 0.447·20-s − 1.25·23-s + 1.51·28-s + 0.557·29-s + 1.79·31-s + 0.676·35-s + 2.63·37-s − 0.937·41-s + 0.609·43-s + 1.80·44-s + 0.875·47-s − 2/7·49-s + 1.10·52-s − 0.824·53-s + 0.809·55-s − 1.17·59-s + 0.896·61-s − 64-s + 0.496·65-s − 0.244·67-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.269968890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.269968890\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 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
−10.37802296429431719248202563402, −10.16143255715236096694498936611, −9.463538113949743957476755824073, −9.130161875272418961155897558854, −8.592302065686632975588167916371, −8.323181365297273262509568131784, −7.84503221359509373921561065675, −7.55507865849627031912550530546, −6.70377090797413947112980043411, −6.36209800037672362467075038845, −6.20669072455171087582507040555, −5.94818136233015567114212422462, −4.86305315376714423181502804439, −4.59648242966108528162028824072, −4.09549348528459852481621896174, −3.66166160214414539189139822770, −2.63261284754577781174387588484, −2.26097898184724496658355889148, −1.51536073064056676997525217643, −1.19796066188443671074681822521,
1.19796066188443671074681822521, 1.51536073064056676997525217643, 2.26097898184724496658355889148, 2.63261284754577781174387588484, 3.66166160214414539189139822770, 4.09549348528459852481621896174, 4.59648242966108528162028824072, 4.86305315376714423181502804439, 5.94818136233015567114212422462, 6.20669072455171087582507040555, 6.36209800037672362467075038845, 6.70377090797413947112980043411, 7.55507865849627031912550530546, 7.84503221359509373921561065675, 8.323181365297273262509568131784, 8.592302065686632975588167916371, 9.130161875272418961155897558854, 9.463538113949743957476755824073, 10.16143255715236096694498936611, 10.37802296429431719248202563402
