| L(s) = 1 | − 2·3-s + 8·5-s + 3·9-s − 16·15-s − 4·16-s + 4·23-s + 38·25-s − 4·27-s − 10·31-s + 6·37-s + 24·45-s + 4·47-s + 8·48-s − 13·49-s + 12·53-s − 20·59-s − 2·67-s − 8·69-s − 76·75-s − 32·80-s + 5·81-s + 24·89-s + 20·93-s + 10·97-s − 14·103-s − 12·111-s + 12·113-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3.57·5-s + 9-s − 4.13·15-s − 16-s + 0.834·23-s + 38/5·25-s − 0.769·27-s − 1.79·31-s + 0.986·37-s + 3.57·45-s + 0.583·47-s + 1.15·48-s − 1.85·49-s + 1.64·53-s − 2.60·59-s − 0.244·67-s − 0.963·69-s − 8.77·75-s − 3.57·80-s + 5/9·81-s + 2.54·89-s + 2.07·93-s + 1.01·97-s − 1.37·103-s − 1.13·111-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.153340679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153340679\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−9.598150501746471856910168767253, −8.973480572648402934204682616447, −8.972019134073409502580254099142, −7.79298545455764963624452711488, −6.98676550080062361288693213459, −6.75395532990341575165448572479, −6.18537700457688272303978752564, −5.80045457911231193627474837008, −5.59539380126593624738687647200, −4.79085093103530693850477639954, −4.73167031165690126954598856077, −3.29412467945553736245288804676, −2.38116356225299620011633985249, −1.94003785572390378144466057337, −1.22560267668749840643223616905,
1.22560267668749840643223616905, 1.94003785572390378144466057337, 2.38116356225299620011633985249, 3.29412467945553736245288804676, 4.73167031165690126954598856077, 4.79085093103530693850477639954, 5.59539380126593624738687647200, 5.80045457911231193627474837008, 6.18537700457688272303978752564, 6.75395532990341575165448572479, 6.98676550080062361288693213459, 7.79298545455764963624452711488, 8.972019134073409502580254099142, 8.973480572648402934204682616447, 9.598150501746471856910168767253
