| L(s) = 1 | − 2·4-s − 3·13-s + 3·16-s − 25-s − 37-s − 43-s + 6·52-s − 4·64-s − 2·67-s − 2·79-s − 3·97-s + 2·100-s − 3·103-s − 109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 2·172-s + ⋯ |
| L(s) = 1 | − 2·4-s − 3·13-s + 3·16-s − 25-s − 37-s − 43-s + 6·52-s − 4·64-s − 2·67-s − 2·79-s − 3·97-s + 2·100-s − 3·103-s − 109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 2·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09980103729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09980103729\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
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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
−8.874704509138622740975186719162, −8.320087252737047204775071169508, −8.257049266024404746902099561890, −7.75438943832620766636523381249, −7.48791100877396169877378558219, −6.97759157012361549302620265784, −6.87314913958770415880115059590, −6.06205324974830449028876963511, −5.64151934250159073352078862002, −5.26684051917615546558072143272, −5.18834549104157655629253304149, −4.49275503056605956887332218662, −4.48396005193632839091050347703, −4.01482350365401495404561296657, −3.50975103312707165147560332916, −2.82359833109246826892104935142, −2.74931456438650015158566341570, −1.83539030818223934807081725093, −1.37018080199409848013997780386, −0.18576498621895713272144128640,
0.18576498621895713272144128640, 1.37018080199409848013997780386, 1.83539030818223934807081725093, 2.74931456438650015158566341570, 2.82359833109246826892104935142, 3.50975103312707165147560332916, 4.01482350365401495404561296657, 4.48396005193632839091050347703, 4.49275503056605956887332218662, 5.18834549104157655629253304149, 5.26684051917615546558072143272, 5.64151934250159073352078862002, 6.06205324974830449028876963511, 6.87314913958770415880115059590, 6.97759157012361549302620265784, 7.48791100877396169877378558219, 7.75438943832620766636523381249, 8.257049266024404746902099561890, 8.320087252737047204775071169508, 8.874704509138622740975186719162
