| L(s) = 1 | − 2·2-s − 12·5-s + 7·7-s + 8·8-s + 24·10-s − 60·11-s + 79·13-s − 14·14-s − 16·16-s − 216·17-s + 22·19-s + 120·22-s + 132·23-s + 125·25-s − 158·26-s − 96·29-s − 20·31-s + 432·34-s − 84·35-s − 338·37-s − 44·38-s − 96·40-s − 192·41-s − 488·43-s − 264·46-s − 204·47-s + 343·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.07·5-s + 0.377·7-s + 0.353·8-s + 0.758·10-s − 1.64·11-s + 1.68·13-s − 0.267·14-s − 1/4·16-s − 3.08·17-s + 0.265·19-s + 1.16·22-s + 1.19·23-s + 25-s − 1.19·26-s − 0.614·29-s − 0.115·31-s + 2.17·34-s − 0.405·35-s − 1.50·37-s − 0.187·38-s − 0.379·40-s − 0.731·41-s − 1.73·43-s − 0.846·46-s − 0.633·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4067355109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4067355109\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 12 T + 19 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 p T + p^{3} T^{2} )( 1 + 4 p T + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 60 T + 2269 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 79 T + 4044 T^{2} - 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 108 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 132 T + 5257 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 96 T - 15173 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 20 T - 29391 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 169 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 192 T - 32057 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 488 T + 158637 T^{2} + 488 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 204 T - 62207 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 156 T - 181043 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 83 T - 220092 T^{2} + 83 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 47 T - 298554 T^{2} + 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 216 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 p T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 529 T - 213198 T^{2} - 529 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1128 T + 700597 T^{2} - 1128 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 605 T - 546648 T^{2} + 605 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.94647966920882713954186003787, −11.76414154521276121845395968847, −11.68509704451768316935694647811, −10.83712016222700429515281777126, −10.75457685480699446219214077685, −10.50268907966505490694944889522, −9.406382352617862254972327150969, −8.712024713422528983467342441716, −8.709295051818692957249046152001, −8.173926867466297002594354530503, −7.56409575941478702958585898153, −6.79351852811749482714866076853, −6.67627627889019265633997145464, −5.40909217268660465479848964951, −4.98660819558090645933975044450, −4.24312075181604900193358279435, −3.58948485079848370333867939823, −2.64695539969251581289874291324, −1.65110844180796473436661590595, −0.33754000999539810713039173996,
0.33754000999539810713039173996, 1.65110844180796473436661590595, 2.64695539969251581289874291324, 3.58948485079848370333867939823, 4.24312075181604900193358279435, 4.98660819558090645933975044450, 5.40909217268660465479848964951, 6.67627627889019265633997145464, 6.79351852811749482714866076853, 7.56409575941478702958585898153, 8.173926867466297002594354530503, 8.709295051818692957249046152001, 8.712024713422528983467342441716, 9.406382352617862254972327150969, 10.50268907966505490694944889522, 10.75457685480699446219214077685, 10.83712016222700429515281777126, 11.68509704451768316935694647811, 11.76414154521276121845395968847, 12.94647966920882713954186003787
