| L(s) = 1 | + 3-s − 5-s + 3·9-s − 6·11-s − 4·13-s − 15-s − 6·17-s + 8·19-s − 3·23-s + 8·27-s + 6·29-s + 2·31-s − 6·33-s − 8·37-s − 4·39-s + 6·41-s + 10·43-s − 3·45-s − 6·51-s − 12·53-s + 6·55-s + 8·57-s − 61-s + 4·65-s + 7·67-s − 3·69-s − 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 9-s − 1.80·11-s − 1.10·13-s − 0.258·15-s − 1.45·17-s + 1.83·19-s − 0.625·23-s + 1.53·27-s + 1.11·29-s + 0.359·31-s − 1.04·33-s − 1.31·37-s − 0.640·39-s + 0.937·41-s + 1.52·43-s − 0.447·45-s − 0.840·51-s − 1.64·53-s + 0.809·55-s + 1.05·57-s − 0.128·61-s + 0.496·65-s + 0.855·67-s − 0.361·69-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.624585999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.624585999\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.22665326279681432802530989079, −9.750863008896782505080634156278, −9.364108686856853557880373933773, −9.113886758065741616957605798370, −8.281360903314058443916661374250, −8.188250272607580126965554398450, −7.74296214996388938098857911198, −7.27933364803808815766743741765, −7.03918237554094277966136532677, −6.57048773898033294378105196502, −5.83499383544825359569944306323, −5.34470118528749854473689820698, −4.80882369970073147010761300571, −4.57592037970624273610771017081, −4.06483709625315044635575586511, −3.23958708758297670858193642353, −2.77635348021519009364105900721, −2.46747832701378377104507718087, −1.64973143659609205796173223313, −0.56245545339233962096906289444,
0.56245545339233962096906289444, 1.64973143659609205796173223313, 2.46747832701378377104507718087, 2.77635348021519009364105900721, 3.23958708758297670858193642353, 4.06483709625315044635575586511, 4.57592037970624273610771017081, 4.80882369970073147010761300571, 5.34470118528749854473689820698, 5.83499383544825359569944306323, 6.57048773898033294378105196502, 7.03918237554094277966136532677, 7.27933364803808815766743741765, 7.74296214996388938098857911198, 8.188250272607580126965554398450, 8.281360903314058443916661374250, 9.113886758065741616957605798370, 9.364108686856853557880373933773, 9.750863008896782505080634156278, 10.22665326279681432802530989079
