| L(s) = 1 | + 5·4-s + 9·9-s − 6·11-s + 16·16-s − 44·19-s − 156·29-s − 64·31-s + 45·36-s − 42·41-s − 30·44-s − 94·49-s − 174·59-s − 112·61-s + 115·64-s − 220·76-s + 76·79-s − 54·99-s + 156·101-s + 208·109-s − 780·116-s − 221·121-s − 320·124-s + 127-s + 131-s + 137-s + 139-s + 144·144-s + ⋯ |
| L(s) = 1 | + 5/4·4-s + 9-s − 0.545·11-s + 16-s − 2.31·19-s − 5.37·29-s − 2.06·31-s + 5/4·36-s − 1.02·41-s − 0.681·44-s − 1.91·49-s − 2.94·59-s − 1.83·61-s + 1.79·64-s − 2.89·76-s + 0.962·79-s − 0.545·99-s + 1.54·101-s + 1.90·109-s − 6.72·116-s − 1.82·121-s − 2.58·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02527408230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02527408230\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 5 T^{2} + 9 T^{4} - 5 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 23 T^{2} + p^{4} T^{4} )( 1 + 71 T^{2} + p^{4} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T + 124 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 322 T^{2} + 75123 T^{4} + 322 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 335 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 290 T^{2} - 195741 T^{4} - 290 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 78 T + 2869 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 32 T + 63 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1582 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 21 T + 1828 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 3214 T^{2} + p^{4} T^{4} )( 1 + 3191 T^{2} + p^{4} T^{4} ) \) |
| 47 | $C_2^3$ | \( 1 - 2066 T^{2} - 611325 T^{4} - 2066 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 87 T + 6004 T^{2} + 87 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 56 T - 585 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 8017 T^{2} + 44121168 T^{4} + 8017 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 9110 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 6433 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 38 T - 4797 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 11426 T^{2} + 83095155 T^{4} - 11426 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 5593 T^{2} - 57247632 T^{4} + 5593 p^{4} T^{6} + p^{8} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.876893316501291992598272379763, −8.367412209557000427900939142448, −8.023993351966481392394246726807, −7.84107158532228077503217023619, −7.57271654216396163357351416880, −7.26124969701482941042838466002, −7.25120640805348430337715907471, −6.93069383049080198727463390826, −6.57320728183164972682665553067, −6.08688779296623637405323482038, −6.08271615814626508103784939557, −5.83038551952258134801018397554, −5.42163778318167423634417330016, −5.09372193097662991698157150632, −4.74118804692537614496891167439, −4.39270447797196656570702721761, −4.00897185864602935787592409442, −3.65428756780633052740021703406, −3.28845441195997867013915150294, −3.23806770497025093020722244851, −2.28962656638144170372561036126, −1.95871577414247514224491531714, −1.82650018701440605104984158172, −1.59212591242061573244420113928, −0.03739152028739297915017751937,
0.03739152028739297915017751937, 1.59212591242061573244420113928, 1.82650018701440605104984158172, 1.95871577414247514224491531714, 2.28962656638144170372561036126, 3.23806770497025093020722244851, 3.28845441195997867013915150294, 3.65428756780633052740021703406, 4.00897185864602935787592409442, 4.39270447797196656570702721761, 4.74118804692537614496891167439, 5.09372193097662991698157150632, 5.42163778318167423634417330016, 5.83038551952258134801018397554, 6.08271615814626508103784939557, 6.08688779296623637405323482038, 6.57320728183164972682665553067, 6.93069383049080198727463390826, 7.25120640805348430337715907471, 7.26124969701482941042838466002, 7.57271654216396163357351416880, 7.84107158532228077503217023619, 8.023993351966481392394246726807, 8.367412209557000427900939142448, 8.876893316501291992598272379763
