| L(s) = 1 | − 4·4-s − 7-s − 5·9-s + 6·11-s + 12·16-s + 12·23-s − 10·25-s + 4·28-s − 12·29-s + 20·36-s + 2·37-s + 16·43-s − 24·44-s − 6·49-s − 6·53-s + 5·63-s − 32·64-s − 8·67-s − 30·71-s − 6·77-s − 20·79-s + 16·81-s − 48·92-s − 30·99-s + 40·100-s + 24·107-s + 4·109-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.377·7-s − 5/3·9-s + 1.80·11-s + 3·16-s + 2.50·23-s − 2·25-s + 0.755·28-s − 2.22·29-s + 10/3·36-s + 0.328·37-s + 2.43·43-s − 3.61·44-s − 6/7·49-s − 0.824·53-s + 0.629·63-s − 4·64-s − 0.977·67-s − 3.56·71-s − 0.683·77-s − 2.25·79-s + 16/9·81-s − 5.00·92-s − 3.01·99-s + 4·100-s + 2.32·107-s + 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−9.304264082512726535347659971465, −9.032345851694468357832029856913, −9.024220761448677623128021512222, −8.328654496275509376474273639773, −7.59911177067371416247669368567, −7.26635425778796924944718267659, −6.10623466338338256757564888425, −5.88099830216512306636961685790, −5.44973416215471530225317007593, −4.59603745135488779054730965561, −4.11544018332675436424095243438, −3.50910294340479626549928321873, −2.96661518607538701123679257760, −1.35226351470929278471090971392, 0,
1.35226351470929278471090971392, 2.96661518607538701123679257760, 3.50910294340479626549928321873, 4.11544018332675436424095243438, 4.59603745135488779054730965561, 5.44973416215471530225317007593, 5.88099830216512306636961685790, 6.10623466338338256757564888425, 7.26635425778796924944718267659, 7.59911177067371416247669368567, 8.328654496275509376474273639773, 9.024220761448677623128021512222, 9.032345851694468357832029856913, 9.304264082512726535347659971465
