| L(s) = 1 | + 3·5-s + 7-s − 11-s − 8·13-s + 4·17-s + 8·23-s + 5·25-s + 14·29-s − 11·31-s + 3·35-s − 4·37-s + 8·41-s − 4·43-s − 2·47-s − 6·49-s − 11·53-s − 3·55-s + 7·59-s − 10·61-s − 24·65-s − 10·67-s − 12·71-s + 6·73-s − 77-s − 11·79-s − 22·83-s + 12·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s − 0.301·11-s − 2.21·13-s + 0.970·17-s + 1.66·23-s + 25-s + 2.59·29-s − 1.97·31-s + 0.507·35-s − 0.657·37-s + 1.24·41-s − 0.609·43-s − 0.291·47-s − 6/7·49-s − 1.51·53-s − 0.404·55-s + 0.911·59-s − 1.28·61-s − 2.97·65-s − 1.22·67-s − 1.42·71-s + 0.702·73-s − 0.113·77-s − 1.23·79-s − 2.41·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.674822304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.674822304\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−9.448672979203630271741082129894, −8.854755431610770690849582636258, −8.825790505456950115006492079970, −8.184734481214841461052005947870, −7.61734822815250132847842607840, −7.39089583952878073543513655830, −7.14018844687413372843643723201, −6.47802843432224719329602067326, −6.29798030149334264566211691806, −5.57028933533279822657902450197, −5.37817082947742732822112948661, −4.87084254526608595186970851402, −4.76355513577118630879311214882, −4.19387527583741311891579361334, −3.20629836787024761880809444655, −2.83992034528009289727692786305, −2.72796646846321541373372023178, −1.70348918249731420977608243978, −1.63272246186770323194874655328, −0.57146776508138604861323864386,
0.57146776508138604861323864386, 1.63272246186770323194874655328, 1.70348918249731420977608243978, 2.72796646846321541373372023178, 2.83992034528009289727692786305, 3.20629836787024761880809444655, 4.19387527583741311891579361334, 4.76355513577118630879311214882, 4.87084254526608595186970851402, 5.37817082947742732822112948661, 5.57028933533279822657902450197, 6.29798030149334264566211691806, 6.47802843432224719329602067326, 7.14018844687413372843643723201, 7.39089583952878073543513655830, 7.61734822815250132847842607840, 8.184734481214841461052005947870, 8.825790505456950115006492079970, 8.854755431610770690849582636258, 9.448672979203630271741082129894
