| L(s) = 1 | + 6·4-s − 90·7-s − 2.09e3·13-s − 988·16-s + 2.31e3·19-s − 540·28-s + 7.26e3·31-s − 6.22e3·37-s + 2.07e4·43-s − 2.75e4·49-s − 1.25e4·52-s − 1.52e4·61-s − 1.20e4·64-s − 1.00e5·67-s − 1.49e5·73-s + 1.39e4·76-s + 7.66e4·79-s + 1.88e5·91-s + 1.43e5·97-s + 6.76e4·103-s − 2.28e5·109-s + 8.89e4·112-s − 2.10e5·121-s + 4.35e4·124-s + 127-s + 131-s − 2.08e5·133-s + ⋯ |
| L(s) = 1 | + 3/16·4-s − 0.694·7-s − 3.42·13-s − 0.964·16-s + 1.47·19-s − 0.130·28-s + 1.35·31-s − 0.746·37-s + 1.70·43-s − 1.63·49-s − 0.643·52-s − 0.523·61-s − 0.368·64-s − 2.74·67-s − 3.28·73-s + 0.276·76-s + 1.38·79-s + 2.38·91-s + 1.54·97-s + 0.628·103-s − 1.83·109-s + 0.669·112-s − 1.30·121-s + 0.254·124-s + 5.50e−6·127-s + 5.09e−6·131-s − 1.02·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 45 T + p^{5} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 210102 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 1045 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1222434 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 61 p T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1682834 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 27470298 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3633 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3110 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 82895598 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10355 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 275888094 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 736930506 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 601496598 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7613 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 50445 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 2896613298 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 74710 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 38316 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 6600582406 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3347081102 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 71755 T + p^{5} T^{2} )^{2} \) |
| 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
−11.26193556937044919801696363946, −10.43628332977152029275584423319, −10.07229732276547028347304253400, −9.779183332270461136974721574226, −9.157871165006419855398568979358, −9.004118216724519847416156084935, −7.88189429013987776509786084104, −7.45314690111052363913801440029, −7.28910040037444417653628353035, −6.62211747726913765869515085152, −6.03634568535001505771828619323, −5.26625775222954581518111285767, −4.73079121027576404054921797681, −4.45217118646795709438327832844, −3.26265524999893725840030717542, −2.72103061764649507204445151342, −2.35015981367577130817835094022, −1.28723335102682685158186217578, 0, 0,
1.28723335102682685158186217578, 2.35015981367577130817835094022, 2.72103061764649507204445151342, 3.26265524999893725840030717542, 4.45217118646795709438327832844, 4.73079121027576404054921797681, 5.26625775222954581518111285767, 6.03634568535001505771828619323, 6.62211747726913765869515085152, 7.28910040037444417653628353035, 7.45314690111052363913801440029, 7.88189429013987776509786084104, 9.004118216724519847416156084935, 9.157871165006419855398568979358, 9.779183332270461136974721574226, 10.07229732276547028347304253400, 10.43628332977152029275584423319, 11.26193556937044919801696363946
