| L(s) = 1 | − 8·2-s + 48·4-s + 7·5-s − 256·8-s − 56·10-s + 19·11-s + 889·13-s + 1.28e3·16-s + 1.69e3·17-s − 1.89e3·19-s + 336·20-s − 152·22-s + 5.32e3·23-s − 1.85e3·25-s − 7.11e3·26-s − 2.12e3·29-s − 5.26e3·31-s − 6.14e3·32-s − 1.35e4·34-s − 7.03e3·37-s + 1.51e4·38-s − 1.79e3·40-s + 1.99e4·41-s + 5.76e3·43-s + 912·44-s − 4.26e4·46-s − 1.78e4·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.125·5-s − 1.41·8-s − 0.177·10-s + 0.0473·11-s + 1.45·13-s + 5/4·16-s + 1.42·17-s − 1.20·19-s + 0.187·20-s − 0.0669·22-s + 2.09·23-s − 0.594·25-s − 2.06·26-s − 0.469·29-s − 0.983·31-s − 1.06·32-s − 2.01·34-s − 0.844·37-s + 1.70·38-s − 0.177·40-s + 1.85·41-s + 0.475·43-s + 0.0710·44-s − 2.96·46-s − 1.18·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.566167974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.566167974\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 7 T + 1906 T^{2} - 7 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 19 T + 28514 p T^{2} - 19 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 889 T + 847988 T^{2} - 889 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1694 T + 2268370 T^{2} - 1694 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1897 T + 3704916 T^{2} + 1897 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5326 T + 19110430 T^{2} - 5326 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2125 T + 6077098 T^{2} + 2125 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5264 T + 42840101 T^{2} + 5264 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7031 T + 117158340 T^{2} + 7031 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 19992 T + 326714386 T^{2} - 19992 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5767 T - 85040406 T^{2} - 5767 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 17892 T + 535684798 T^{2} + 17892 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4545 T + 232871938 T^{2} + 4545 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 46333 T + 1965656926 T^{2} - 46333 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6482 T + 615855410 T^{2} + 6482 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1605 p T + 5586677536 T^{2} - 1605 p^{6} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9184 T + 2473261198 T^{2} - 9184 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 52899 T + 3523157380 T^{2} + 52899 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 45838 T + 5621470031 T^{2} + 45838 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 27055 T + 4467920878 T^{2} + 27055 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 234738 T + 24910172242 T^{2} - 234738 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 105777 T + 18591077740 T^{2} + 105777 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.339938621227430688907895362526, −9.250433686810108272879629839697, −8.859633222525651770060216581104, −8.352561494592379815235366178885, −8.025216063259540723066578375934, −7.67341133984996931410735476185, −6.95079956672233518375966177881, −6.92423043031147542356210266854, −6.22053119084316426023025398982, −5.87231382433019196370682268811, −5.38493076642996283714193443668, −4.94904047748663521058253423902, −3.88966941181777696515039336404, −3.79009890096843854618373682343, −3.08098457407957899179468957092, −2.58609905575593744696344641540, −1.81048701400791781482504024887, −1.51314135574591604801790441557, −0.71721939164042990030226893847, −0.59622393395078524202151343740,
0.59622393395078524202151343740, 0.71721939164042990030226893847, 1.51314135574591604801790441557, 1.81048701400791781482504024887, 2.58609905575593744696344641540, 3.08098457407957899179468957092, 3.79009890096843854618373682343, 3.88966941181777696515039336404, 4.94904047748663521058253423902, 5.38493076642996283714193443668, 5.87231382433019196370682268811, 6.22053119084316426023025398982, 6.92423043031147542356210266854, 6.95079956672233518375966177881, 7.67341133984996931410735476185, 8.025216063259540723066578375934, 8.352561494592379815235366178885, 8.859633222525651770060216581104, 9.250433686810108272879629839697, 9.339938621227430688907895362526
