| L(s) = 1 | + 4·9-s + 12·13-s + 4·17-s + 8·29-s + 4·37-s − 4·49-s + 12·53-s − 4·61-s − 28·73-s + 7·81-s − 20·89-s − 4·97-s + 20·109-s − 12·113-s + 48·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + 157-s + 163-s + 167-s + 82·169-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 3.32·13-s + 0.970·17-s + 1.48·29-s + 0.657·37-s − 4/7·49-s + 1.64·53-s − 0.512·61-s − 3.27·73-s + 7/9·81-s − 2.11·89-s − 0.406·97-s + 1.91·109-s − 1.12·113-s + 4.43·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.29·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-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
| \(L(1)\) |
\(\approx\) |
\(5.564691267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.564691267\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 156 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−8.250552925698858575300674501627, −7.82494215893329287215408386478, −7.58906612239762773616690941839, −7.02193567588646690256612669422, −6.84370338766226252062203870778, −6.43574341329287110865412534403, −5.98925186658998332068467220806, −5.86928428854149287757979677308, −5.52630670013222263932221186833, −4.90219438092375263260437044660, −4.46759582137381515066940398503, −4.18742602701225310349503185721, −3.78843223977094485992626695654, −3.57903464169436938992632957566, −2.91425034053346015103178943160, −2.78325390232295681405329267640, −1.82418505940808315899384319502, −1.30273997638080493822569563182, −1.30214997765394643142706973938, −0.67027293343512682116127433632,
0.67027293343512682116127433632, 1.30214997765394643142706973938, 1.30273997638080493822569563182, 1.82418505940808315899384319502, 2.78325390232295681405329267640, 2.91425034053346015103178943160, 3.57903464169436938992632957566, 3.78843223977094485992626695654, 4.18742602701225310349503185721, 4.46759582137381515066940398503, 4.90219438092375263260437044660, 5.52630670013222263932221186833, 5.86928428854149287757979677308, 5.98925186658998332068467220806, 6.43574341329287110865412534403, 6.84370338766226252062203870778, 7.02193567588646690256612669422, 7.58906612239762773616690941839, 7.82494215893329287215408386478, 8.250552925698858575300674501627
