| L(s) = 1 | − 2·5-s − 2·7-s − 32·11-s + 27·13-s + 22·17-s + 38·19-s + 40·23-s + 25·25-s − 30·29-s + 4·35-s + 24·41-s + 49·43-s + 46·47-s − 95·49-s + 84·53-s + 64·55-s + 114·59-s + 97·61-s − 54·65-s − 45·67-s + 84·71-s − 35·73-s + 64·77-s − 153·79-s + 292·83-s − 44·85-s + 66·89-s + ⋯ |
| L(s) = 1 | − 2/5·5-s − 2/7·7-s − 2.90·11-s + 2.07·13-s + 1.29·17-s + 2·19-s + 1.73·23-s + 25-s − 1.03·29-s + 4/35·35-s + 0.585·41-s + 1.13·43-s + 0.978·47-s − 1.93·49-s + 1.58·53-s + 1.16·55-s + 1.93·59-s + 1.59·61-s − 0.830·65-s − 0.671·67-s + 1.18·71-s − 0.479·73-s + 0.831·77-s − 1.93·79-s + 3.51·83-s − 0.517·85-s + 0.741·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.654201107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.654201107\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 27 T + 412 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 22 T + 195 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 40 T + 1071 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 30 T + 1141 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 601 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2495 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 24 T + 1873 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 49 T + 552 T^{2} - 49 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 46 T - 93 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 84 T + 5161 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T + 7813 T^{2} - 114 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 97 T + 5688 T^{2} - 97 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 45 T + 5164 T^{2} + 45 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 84 T + 7393 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 35 T - 4104 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 66 T + 9373 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 108 T + 13297 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.47581565875536462198855790007, −10.29008753756581841499783632647, −9.538194934722575689320531135803, −9.312121052167459808469163432601, −8.643813702587868256826476155281, −8.314700291319893249103639520274, −7.81859307845830652290363295572, −7.51307232752556592056700426166, −7.17540988824614920469543831922, −6.56661383186208885763347022746, −5.68843933235621825895167758185, −5.55687276631341101514144957769, −5.22792365114790030580674511415, −4.68046421249714392658196457289, −3.60824068578796079756010863262, −3.48602316821814774281724033308, −2.88516715063849563599291067144, −2.36268518168293775311217291288, −1.04997781685056742268644035198, −0.74717614808431635741248711236,
0.74717614808431635741248711236, 1.04997781685056742268644035198, 2.36268518168293775311217291288, 2.88516715063849563599291067144, 3.48602316821814774281724033308, 3.60824068578796079756010863262, 4.68046421249714392658196457289, 5.22792365114790030580674511415, 5.55687276631341101514144957769, 5.68843933235621825895167758185, 6.56661383186208885763347022746, 7.17540988824614920469543831922, 7.51307232752556592056700426166, 7.81859307845830652290363295572, 8.314700291319893249103639520274, 8.643813702587868256826476155281, 9.312121052167459808469163432601, 9.538194934722575689320531135803, 10.29008753756581841499783632647, 10.47581565875536462198855790007
