L(s) = 1 | − 2-s + 2·4-s + 5-s − 5·8-s − 10-s − 4·11-s + 2·13-s + 5·16-s − 4·17-s + 8·19-s + 2·20-s + 4·22-s − 2·26-s − 2·29-s − 10·32-s + 4·34-s − 20·37-s − 8·38-s − 5·40-s + 10·41-s − 4·43-s − 8·44-s + 8·47-s + 7·49-s + 4·52-s + 20·53-s − 4·55-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 4-s + 0.447·5-s − 1.76·8-s − 0.316·10-s − 1.20·11-s + 0.554·13-s + 5/4·16-s − 0.970·17-s + 1.83·19-s + 0.447·20-s + 0.852·22-s − 0.392·26-s − 0.371·29-s − 1.76·32-s + 0.685·34-s − 3.28·37-s − 1.29·38-s − 0.790·40-s + 1.56·41-s − 0.609·43-s − 1.20·44-s + 1.16·47-s + 49-s + 0.554·52-s + 2.74·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.132439658\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.132439658\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−11.63631279952971224303853714579, −10.77963890822316167601103086561, −10.64152928560301242275577443534, −10.20107499208583961364476901053, −9.625966598251987688812605038515, −9.121950071644673854643705551091, −8.810133370240374653294235948282, −8.474299849568466977957851395955, −7.70207052092179351489798616263, −7.18235048403042326045897491627, −7.08021719328771249134578233951, −6.19478176171239042880879059654, −5.88842810105885743785816001124, −5.30528964311774442667302699047, −4.94890624958160086104752971198, −3.54463189125455891335815274182, −3.47803764713987674617217746521, −2.32517192390184871962881805736, −2.20696536058656135890082817817, −0.76517041338047048148171756520,
0.76517041338047048148171756520, 2.20696536058656135890082817817, 2.32517192390184871962881805736, 3.47803764713987674617217746521, 3.54463189125455891335815274182, 4.94890624958160086104752971198, 5.30528964311774442667302699047, 5.88842810105885743785816001124, 6.19478176171239042880879059654, 7.08021719328771249134578233951, 7.18235048403042326045897491627, 7.70207052092179351489798616263, 8.474299849568466977957851395955, 8.810133370240374653294235948282, 9.121950071644673854643705551091, 9.625966598251987688812605038515, 10.20107499208583961364476901053, 10.64152928560301242275577443534, 10.77963890822316167601103086561, 11.63631279952971224303853714579