| L(s) = 1 | − 4·4-s − 2·7-s + 9-s − 10·11-s + 12·16-s − 2·23-s − 6·25-s + 8·28-s − 4·36-s + 16·37-s − 2·43-s + 40·44-s − 3·49-s + 6·53-s − 2·63-s − 32·64-s − 30·67-s − 28·71-s + 20·77-s − 32·79-s + 81-s + 8·92-s − 10·99-s + 24·100-s + 22·109-s − 24·112-s − 8·113-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.755·7-s + 1/3·9-s − 3.01·11-s + 3·16-s − 0.417·23-s − 6/5·25-s + 1.51·28-s − 2/3·36-s + 2.63·37-s − 0.304·43-s + 6.03·44-s − 3/7·49-s + 0.824·53-s − 0.251·63-s − 4·64-s − 3.66·67-s − 3.32·71-s + 2.27·77-s − 3.60·79-s + 1/9·81-s + 0.834·92-s − 1.00·99-s + 12/5·100-s + 2.10·109-s − 2.26·112-s − 0.752·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 815409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 815409 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
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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
−7.73502482152706926438296491692, −7.51672561268763905855633948060, −7.29191481743015083437539167548, −5.96338923144874103909303632728, −5.81908606252366345509022093604, −5.75522294186426097256212673359, −4.74387622427760402111791858890, −4.66504278434376881732923135220, −4.24182954023649007385411626326, −3.47776319336265678852062283368, −2.92856704948279397641421895416, −2.54707406990528808772182221251, −1.34730029274708816959785017632, 0, 0,
1.34730029274708816959785017632, 2.54707406990528808772182221251, 2.92856704948279397641421895416, 3.47776319336265678852062283368, 4.24182954023649007385411626326, 4.66504278434376881732923135220, 4.74387622427760402111791858890, 5.75522294186426097256212673359, 5.81908606252366345509022093604, 5.96338923144874103909303632728, 7.29191481743015083437539167548, 7.51672561268763905855633948060, 7.73502482152706926438296491692
