| L(s) = 1 | + 6·3-s + 2·4-s − 135·9-s + 22·11-s + 12·12-s − 252·16-s − 554·23-s − 1.35e3·27-s − 2.72e3·31-s + 132·33-s − 270·36-s − 334·37-s + 44·44-s − 3.40e3·47-s − 1.51e3·48-s + 1.80e3·49-s − 9.04e3·53-s − 4.72e3·59-s − 1.01e3·64-s + 5.60e3·67-s − 3.32e3·69-s + 6.79e3·71-s + 1.13e4·81-s − 9.34e3·89-s − 1.10e3·92-s − 1.63e4·93-s − 8.49e3·97-s + ⋯ |
| L(s) = 1 | + 2/3·3-s + 1/8·4-s − 5/3·9-s + 2/11·11-s + 1/12·12-s − 0.984·16-s − 1.04·23-s − 1.85·27-s − 2.83·31-s + 4/33·33-s − 0.208·36-s − 0.243·37-s + 1/44·44-s − 1.54·47-s − 0.656·48-s + 0.750·49-s − 3.21·53-s − 1.35·59-s − 0.248·64-s + 1.24·67-s − 0.698·69-s + 1.34·71-s + 1.72·81-s − 1.17·89-s − 0.130·92-s − 1.89·93-s − 0.902·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.002897582140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002897582140\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 2 p T + p^{4} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{8} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 1802 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22442 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 114122 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 250922 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 277 T + p^{4} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 38 p T + p^{4} T^{2} )( 1 + 38 p T + p^{4} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 1363 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 167 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4522442 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 5385602 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 1702 T + p^{4} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4522 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 2363 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 11966402 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2803 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3397 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 45779402 T^{2} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 40803842 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 94223522 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4673 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4247 T + p^{4} T^{2} )^{2} \) |
| 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
−11.89162924533053286219221545766, −11.06158821509574026868433839292, −10.86161173707675685694342217228, −9.704282094549909664031461304365, −9.651958051229868550050956379478, −8.914891303963064000220213464252, −8.780054496468348791137922427550, −8.069398679109395188051646466640, −7.75694562905444240463841957923, −7.17640116778495115305060992367, −6.26494326449777664720647207203, −6.22460089261279439438049104874, −5.31048023468580542889728320341, −5.00175978938827614485530661738, −3.98197714711368249193029962393, −3.49756763733772840259963450049, −2.91475869428826658556062734336, −2.16427742903023234263273454480, −1.66780884597158018979410253339, −0.01501867936361110857932038334,
0.01501867936361110857932038334, 1.66780884597158018979410253339, 2.16427742903023234263273454480, 2.91475869428826658556062734336, 3.49756763733772840259963450049, 3.98197714711368249193029962393, 5.00175978938827614485530661738, 5.31048023468580542889728320341, 6.22460089261279439438049104874, 6.26494326449777664720647207203, 7.17640116778495115305060992367, 7.75694562905444240463841957923, 8.069398679109395188051646466640, 8.780054496468348791137922427550, 8.914891303963064000220213464252, 9.651958051229868550050956379478, 9.704282094549909664031461304365, 10.86161173707675685694342217228, 11.06158821509574026868433839292, 11.89162924533053286219221545766
