| L(s) = 1 | + 8·2-s + 32·4-s − 234·7-s − 1.94e3·11-s + 4.82e3·13-s − 1.87e3·14-s − 1.02e3·16-s + 9.62e3·17-s − 1.55e4·22-s − 2.71e4·23-s + 3.85e4·26-s − 7.48e3·28-s + 5.95e4·31-s − 8.19e3·32-s + 7.69e4·34-s + 3.08e4·37-s − 1.90e5·41-s − 1.97e5·43-s − 6.22e4·44-s − 2.16e5·46-s − 1.33e5·47-s + 2.73e4·49-s + 1.54e5·52-s + 9.69e4·53-s − 2.74e5·61-s + 4.76e5·62-s − 3.27e4·64-s + ⋯ |
| L(s) = 1 | + 2-s + 1/2·4-s − 0.682·7-s − 1.46·11-s + 2.19·13-s − 0.682·14-s − 1/4·16-s + 1.95·17-s − 1.46·22-s − 2.22·23-s + 2.19·26-s − 0.341·28-s + 1.99·31-s − 1/4·32-s + 1.95·34-s + 0.608·37-s − 2.77·41-s − 2.48·43-s − 0.730·44-s − 2.22·46-s − 1.28·47-s + 0.232·49-s + 1.09·52-s + 0.651·53-s − 1.20·61-s + 1.99·62-s − 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3507836153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3507836153\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 - p^{3} T + p^{5} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 234 T + 27378 T^{2} + 234 p^{6} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 972 T + p^{6} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4824 T + 11635488 T^{2} - 4824 p^{6} T^{3} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9624 T + 46310688 T^{2} - 9624 p^{6} T^{3} + p^{12} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 61144162 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 27114 T + 367584498 T^{2} + 27114 p^{6} T^{3} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 881644142 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 29752 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 30816 T + 474812928 T^{2} - 30816 p^{6} T^{3} + p^{12} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 95472 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 197226 T + 19449047538 T^{2} + 197226 p^{6} T^{3} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 133146 T + 8863928658 T^{2} + 133146 p^{6} T^{3} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 96936 T + 4698294048 T^{2} - 96936 p^{6} T^{3} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 63883457282 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 137248 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 375354 T + 70445312658 T^{2} + 375354 p^{6} T^{3} + p^{12} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 255312 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 572184 T + 163697264928 T^{2} - 572184 p^{6} T^{3} + p^{12} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 209848207358 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 906054 T + 410466925458 T^{2} + 906054 p^{6} T^{3} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 770451109022 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 946584 T + 448010634528 T^{2} + 946584 p^{6} T^{3} + p^{12} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.50084229560631427612250220294, −9.882805409547252013374692525740, −9.842124133460237382362878296255, −8.800242263540198136371058317256, −8.357469719615277823032696983049, −7.967431617776658638427834787851, −7.87007500134208947845651630992, −6.77307330313718704866313938437, −6.53862216144350186423832738775, −5.87136416524364103504235732058, −5.82008861589255729123514716186, −5.07731677601586652510082944192, −4.69802688309093282352240900572, −3.72611894427686661979474302495, −3.66579542073105563647941493304, −3.03759829803587836373827769819, −2.62673705131326244982929341378, −1.49741594545339629707173326109, −1.34968405550122650935969286314, −0.10329663209891376298036331743,
0.10329663209891376298036331743, 1.34968405550122650935969286314, 1.49741594545339629707173326109, 2.62673705131326244982929341378, 3.03759829803587836373827769819, 3.66579542073105563647941493304, 3.72611894427686661979474302495, 4.69802688309093282352240900572, 5.07731677601586652510082944192, 5.82008861589255729123514716186, 5.87136416524364103504235732058, 6.53862216144350186423832738775, 6.77307330313718704866313938437, 7.87007500134208947845651630992, 7.967431617776658638427834787851, 8.357469719615277823032696983049, 8.800242263540198136371058317256, 9.842124133460237382362878296255, 9.882805409547252013374692525740, 10.50084229560631427612250220294
