| L(s) = 1 | − 2·2-s + 3·3-s − 5·5-s − 6·6-s − 68·7-s + 8·8-s + 10·10-s − 66·11-s + 16·13-s + 136·14-s − 15·15-s − 16·16-s + 96·17-s − 133·19-s − 204·21-s + 132·22-s + 78·23-s + 24·24-s − 32·26-s − 27·27-s + 225·29-s + 30·30-s + 70·31-s − 198·33-s − 192·34-s + 340·35-s + 256·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s − 3.67·7-s + 0.353·8-s + 0.316·10-s − 1.80·11-s + 0.341·13-s + 2.59·14-s − 0.258·15-s − 1/4·16-s + 1.36·17-s − 1.60·19-s − 2.11·21-s + 1.27·22-s + 0.707·23-s + 0.204·24-s − 0.241·26-s − 0.192·27-s + 1.44·29-s + 0.182·30-s + 0.405·31-s − 1.04·33-s − 0.968·34-s + 1.64·35-s + 1.13·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8084204205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8084204205\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 19 | $C_2$ | \( 1 + 7 p T + p^{3} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 p T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 16 T - 1941 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 96 T + 4303 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 225 T + 26236 T^{2} - 225 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 35 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 128 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 54 T - 66005 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 356 T + 47229 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T - 102923 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 342 T - 31913 T^{2} + 342 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 411 T - 36458 T^{2} - 411 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 835 T + 470244 T^{2} - 835 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 610 T + 71337 T^{2} - 610 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 855 T + 373114 T^{2} - 855 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 712 T + 117927 T^{2} - 712 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 841 T + 214242 T^{2} - 841 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 924 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1437 T + 1360000 T^{2} - 1437 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 250 T - 850173 T^{2} - 250 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.15272117074158021869637377024, −10.12419934785176816018416765416, −9.639621713842460618178920750037, −9.497938571113552147825216421015, −8.634982038678164684586605573869, −8.544769764488129690953306251664, −7.985713052331148317645718984255, −7.55024370132163561578549273662, −6.86483130235819522401793611467, −6.58629323362275500707511254928, −6.24052262682231342829331961795, −5.61965769225545750478784441623, −5.04627224563287881461826171332, −4.18981235900837740692250025853, −3.43469106168511394475692603362, −3.40069008438139682954583498954, −2.62617129881666946407877329895, −2.42483229147245222858628914719, −0.62620738765817852379901827165, −0.51569184080051898963773472955,
0.51569184080051898963773472955, 0.62620738765817852379901827165, 2.42483229147245222858628914719, 2.62617129881666946407877329895, 3.40069008438139682954583498954, 3.43469106168511394475692603362, 4.18981235900837740692250025853, 5.04627224563287881461826171332, 5.61965769225545750478784441623, 6.24052262682231342829331961795, 6.58629323362275500707511254928, 6.86483130235819522401793611467, 7.55024370132163561578549273662, 7.985713052331148317645718984255, 8.544769764488129690953306251664, 8.634982038678164684586605573869, 9.497938571113552147825216421015, 9.639621713842460618178920750037, 10.12419934785176816018416765416, 10.15272117074158021869637377024
