L(s) = 1 | − 4-s − 9-s − 12·11-s + 16-s − 12·29-s + 16·31-s + 36-s + 20·41-s + 12·44-s + 10·49-s + 24·59-s + 8·61-s − 64-s + 12·79-s + 81-s + 12·99-s − 12·101-s + 32·109-s + 12·116-s + 86·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1/3·9-s − 3.61·11-s + 1/4·16-s − 2.22·29-s + 2.87·31-s + 1/6·36-s + 3.12·41-s + 1.80·44-s + 10/7·49-s + 3.12·59-s + 1.02·61-s − 1/8·64-s + 1.35·79-s + 1/9·81-s + 1.20·99-s − 1.19·101-s + 3.06·109-s + 1.11·116-s + 7.81·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.624082333\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.624082333\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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
−8.591167575129918791063899175492, −8.415613214488735574156690846074, −7.901102949239671674781406277249, −7.84265309185275044634081505004, −7.32227968199930107805548669746, −7.26140892364201790955077972701, −6.38040414684730078473907768136, −6.12071835785476599376373508750, −5.50503104335153478734606559426, −5.33329803836054446705019870293, −5.25598346695175173035151017149, −4.59504735090367661197350499922, −4.13565269947342797802947521012, −3.85411211494975631988418870243, −2.93815793600697995641235179366, −2.86451817831585681593443569624, −2.28079514713934646853800232924, −2.13322549172130578241172504284, −0.65906614870420848294484939775, −0.63325439556875178328489974446,
0.63325439556875178328489974446, 0.65906614870420848294484939775, 2.13322549172130578241172504284, 2.28079514713934646853800232924, 2.86451817831585681593443569624, 2.93815793600697995641235179366, 3.85411211494975631988418870243, 4.13565269947342797802947521012, 4.59504735090367661197350499922, 5.25598346695175173035151017149, 5.33329803836054446705019870293, 5.50503104335153478734606559426, 6.12071835785476599376373508750, 6.38040414684730078473907768136, 7.26140892364201790955077972701, 7.32227968199930107805548669746, 7.84265309185275044634081505004, 7.901102949239671674781406277249, 8.415613214488735574156690846074, 8.591167575129918791063899175492