| L(s) = 1 | − 2·3-s − 3·9-s + 6·11-s − 6·17-s − 2·19-s + 14·27-s − 12·33-s + 18·41-s − 8·43-s − 2·49-s + 12·51-s + 4·57-s − 24·59-s − 22·67-s + 14·73-s − 4·81-s − 30·83-s + 6·89-s − 28·97-s − 18·99-s − 18·107-s + 30·113-s + 5·121-s − 36·123-s + 127-s + 16·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 9-s + 1.80·11-s − 1.45·17-s − 0.458·19-s + 2.69·27-s − 2.08·33-s + 2.81·41-s − 1.21·43-s − 2/7·49-s + 1.68·51-s + 0.529·57-s − 3.12·59-s − 2.68·67-s + 1.63·73-s − 4/9·81-s − 3.29·83-s + 0.635·89-s − 2.84·97-s − 1.80·99-s − 1.74·107-s + 2.82·113-s + 5/11·121-s − 3.24·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−7.70335810156829705863391904265, −7.54264307730095337048471741758, −6.89196383428079211455618868444, −6.72442417220664239107111257784, −6.26549578694454350546618644962, −6.18547764499815853950078549552, −5.82447263033605320289712028178, −5.52931567541923221027731900029, −4.91517786982457014928812181189, −4.62676846698900744107986239736, −4.25515486851587908396506549994, −4.05831087742214772749981375137, −3.37140572117864993375075955325, −2.90456731103804175914770569122, −2.66373979462212274946552250522, −2.00416229582808273777504770876, −1.41508994746583727573027361004, −1.02528344775625259459207254184, 0, 0,
1.02528344775625259459207254184, 1.41508994746583727573027361004, 2.00416229582808273777504770876, 2.66373979462212274946552250522, 2.90456731103804175914770569122, 3.37140572117864993375075955325, 4.05831087742214772749981375137, 4.25515486851587908396506549994, 4.62676846698900744107986239736, 4.91517786982457014928812181189, 5.52931567541923221027731900029, 5.82447263033605320289712028178, 6.18547764499815853950078549552, 6.26549578694454350546618644962, 6.72442417220664239107111257784, 6.89196383428079211455618868444, 7.54264307730095337048471741758, 7.70335810156829705863391904265
