| L(s) = 1 | − 4·5-s − 6·7-s + 8·11-s − 5·13-s − 8·19-s − 4·23-s + 5·25-s − 8·29-s + 2·31-s + 24·35-s − 10·37-s − 8·41-s + 5·43-s + 8·47-s + 13·49-s − 4·53-s − 32·55-s − 12·59-s + 61-s + 20·65-s − 3·67-s − 16·71-s + 15·73-s − 48·77-s + 7·79-s + 12·89-s + 30·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 2.26·7-s + 2.41·11-s − 1.38·13-s − 1.83·19-s − 0.834·23-s + 25-s − 1.48·29-s + 0.359·31-s + 4.05·35-s − 1.64·37-s − 1.24·41-s + 0.762·43-s + 1.16·47-s + 13/7·49-s − 0.549·53-s − 4.31·55-s − 1.56·59-s + 0.128·61-s + 2.48·65-s − 0.366·67-s − 1.89·71-s + 1.75·73-s − 5.47·77-s + 0.787·79-s + 1.27·89-s + 3.14·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 16 T + 185 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.26089624552104179897716171551, −9.523345050665575858457372142411, −9.481475078927796904238537000962, −9.197420707841555966961495522464, −8.465510092701553756975477941906, −8.223828357200336120050674113587, −7.40464580214852069383432118590, −7.24624719834599258238111020649, −6.56923399917215595708878143182, −6.53813081810325487917689601453, −6.02931239212541515534791089551, −5.25382324375416410275575658563, −4.26066532879584214213190018881, −4.21729079383030261808581506172, −3.60124396729075636011628585823, −3.43653427898998894136154351412, −2.55793374410957847039359919984, −1.66224303425792408680840645268, 0, 0,
1.66224303425792408680840645268, 2.55793374410957847039359919984, 3.43653427898998894136154351412, 3.60124396729075636011628585823, 4.21729079383030261808581506172, 4.26066532879584214213190018881, 5.25382324375416410275575658563, 6.02931239212541515534791089551, 6.53813081810325487917689601453, 6.56923399917215595708878143182, 7.24624719834599258238111020649, 7.40464580214852069383432118590, 8.223828357200336120050674113587, 8.465510092701553756975477941906, 9.197420707841555966961495522464, 9.481475078927796904238537000962, 9.523345050665575858457372142411, 10.26089624552104179897716171551
