| L(s) = 1 | + 3-s − 2·4-s − 2·9-s − 11-s − 2·12-s − 13-s + 4·16-s − 7·23-s − 6·25-s − 5·27-s − 33-s + 4·36-s − 13·37-s − 39-s + 2·44-s + 10·47-s + 4·48-s − 4·49-s + 2·52-s − 14·59-s − 2·61-s − 8·64-s − 7·69-s + 14·71-s + 5·73-s − 6·75-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 2/3·9-s − 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s − 1.45·23-s − 6/5·25-s − 0.962·27-s − 0.174·33-s + 2/3·36-s − 2.13·37-s − 0.160·39-s + 0.301·44-s + 1.45·47-s + 0.577·48-s − 4/7·49-s + 0.277·52-s − 1.82·59-s − 0.256·61-s − 64-s − 0.842·69-s + 1.66·71-s + 0.585·73-s − 0.692·75-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 409 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.724781306500313564183047768702, −9.088622328388035268404007209626, −8.832432649800327065797774220359, −8.055091941711149295756221945604, −7.954337639673767751663870890031, −7.38458694028301572489883359861, −6.44217399425940761588917568941, −5.94299834416650625993354609877, −5.30149628194833863002070297570, −4.88693569015909842375109321138, −3.82022715714239211736478829647, −3.73276460695406530646140430814, −2.69508532757844159267398654460, −1.84168760951923791424218229463, 0,
1.84168760951923791424218229463, 2.69508532757844159267398654460, 3.73276460695406530646140430814, 3.82022715714239211736478829647, 4.88693569015909842375109321138, 5.30149628194833863002070297570, 5.94299834416650625993354609877, 6.44217399425940761588917568941, 7.38458694028301572489883359861, 7.954337639673767751663870890031, 8.055091941711149295756221945604, 8.832432649800327065797774220359, 9.088622328388035268404007209626, 9.724781306500313564183047768702
