| L(s) = 1 | + 4·3-s + 6·9-s − 6·11-s − 6·17-s + 2·19-s − 7·25-s − 4·27-s − 24·33-s − 12·41-s − 2·43-s − 11·49-s − 24·51-s + 8·57-s − 12·59-s + 8·67-s + 2·73-s − 28·75-s − 37·81-s − 24·83-s + 8·97-s − 36·99-s − 12·107-s − 36·113-s + 5·121-s − 48·123-s + 127-s − 8·129-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s − 1.80·11-s − 1.45·17-s + 0.458·19-s − 7/5·25-s − 0.769·27-s − 4.17·33-s − 1.87·41-s − 0.304·43-s − 1.57·49-s − 3.36·51-s + 1.05·57-s − 1.56·59-s + 0.977·67-s + 0.234·73-s − 3.23·75-s − 4.11·81-s − 2.63·83-s + 0.812·97-s − 3.61·99-s − 1.16·107-s − 3.38·113-s + 5/11·121-s − 4.32·123-s + 0.0887·127-s − 0.704·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−8.074165407662890807330090562647, −7.967601908046062758498727639263, −7.54154771182837907562951747214, −7.28084420153821384462264955127, −6.68402182678737925633599303039, −6.49196180865455768295195716946, −5.75619920146915586569592688283, −5.59503632385320594669237283970, −5.00873351546071334996561317597, −4.79398157420474018498834957177, −4.07981947445611839833999123970, −3.92143101781958611706780236892, −3.20456748057452351457019907730, −3.17174252380299729323959684818, −2.56569723035598843527809779715, −2.43476512807659373600785578284, −1.78284891188202082820909407697, −1.59507462490591482548231560399, 0, 0,
1.59507462490591482548231560399, 1.78284891188202082820909407697, 2.43476512807659373600785578284, 2.56569723035598843527809779715, 3.17174252380299729323959684818, 3.20456748057452351457019907730, 3.92143101781958611706780236892, 4.07981947445611839833999123970, 4.79398157420474018498834957177, 5.00873351546071334996561317597, 5.59503632385320594669237283970, 5.75619920146915586569592688283, 6.49196180865455768295195716946, 6.68402182678737925633599303039, 7.28084420153821384462264955127, 7.54154771182837907562951747214, 7.967601908046062758498727639263, 8.074165407662890807330090562647
