| L(s) = 1 | + 3-s + 4·5-s + 3·9-s − 6·11-s − 2·13-s + 4·15-s − 2·17-s − 19-s + 6·23-s + 5·25-s + 8·27-s + 4·29-s + 20·31-s − 6·33-s + 4·37-s − 2·39-s − 9·41-s − 4·43-s + 12·45-s − 12·47-s − 14·49-s − 2·51-s + 2·53-s − 24·55-s − 57-s − 59-s + 8·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 9-s − 1.80·11-s − 0.554·13-s + 1.03·15-s − 0.485·17-s − 0.229·19-s + 1.25·23-s + 25-s + 1.53·27-s + 0.742·29-s + 3.59·31-s − 1.04·33-s + 0.657·37-s − 0.320·39-s − 1.40·41-s − 0.609·43-s + 1.78·45-s − 1.75·47-s − 2·49-s − 0.280·51-s + 0.274·53-s − 3.23·55-s − 0.132·57-s − 0.130·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.466804212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.466804212\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−11.76258072278365347564656193090, −11.72198242400185260630345459325, −10.62440029208520472919420562422, −10.39000189893979350036112191182, −10.12687790120778233184382470633, −9.636470324861114824405229001086, −9.412106672404616378564180251946, −8.534495941313028896513287988989, −8.079281494323192069601970622651, −7.972884941498347770618588187395, −6.82249317969547774926746422880, −6.64716098032331139238775547717, −6.23671848197268480345670693083, −5.25777917572658162765240638035, −4.76808597674885143024624711630, −4.76325260119392571148515262340, −3.30921564315505916031443927667, −2.62006520481166113612200825421, −2.34711947397844923830619801392, −1.29217626946479508523343033710,
1.29217626946479508523343033710, 2.34711947397844923830619801392, 2.62006520481166113612200825421, 3.30921564315505916031443927667, 4.76325260119392571148515262340, 4.76808597674885143024624711630, 5.25777917572658162765240638035, 6.23671848197268480345670693083, 6.64716098032331139238775547717, 6.82249317969547774926746422880, 7.972884941498347770618588187395, 8.079281494323192069601970622651, 8.534495941313028896513287988989, 9.412106672404616378564180251946, 9.636470324861114824405229001086, 10.12687790120778233184382470633, 10.39000189893979350036112191182, 10.62440029208520472919420562422, 11.72198242400185260630345459325, 11.76258072278365347564656193090
