| L(s) = 1 | − 3-s − 3·11-s + 2·17-s + 25-s + 27-s + 3·33-s − 41-s + 3·43-s − 49-s − 2·51-s − 3·59-s + 3·67-s − 2·73-s − 75-s − 81-s + 4·89-s − 97-s − 2·113-s + 5·121-s + 123-s + 127-s − 3·129-s + 131-s + 137-s + 139-s + 147-s + 149-s + ⋯ |
| L(s) = 1 | − 3-s − 3·11-s + 2·17-s + 25-s + 27-s + 3·33-s − 41-s + 3·43-s − 49-s − 2·51-s − 3·59-s + 3·67-s − 2·73-s − 75-s − 81-s + 4·89-s − 97-s − 2·113-s + 5·121-s + 123-s + 127-s − 3·129-s + 131-s + 137-s + 139-s + 147-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5907089983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5907089983\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 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
−9.315443038155154327358345653219, −9.163812027834394774548878849740, −8.440198218013408817418866236911, −8.076358446923671498948303426255, −7.79211370248316861524361187863, −7.65484550460326826376738680975, −7.10957752643818980496318820883, −6.65135054077798821651233858875, −6.06769170648307973716908360533, −5.76704191023427785652996088699, −5.45497924284220093302993069538, −5.15989707299736134545521197121, −4.74848220071396319869391504358, −4.45583216891991361716697981538, −3.40602800127778600023832834874, −3.27240944795662776286839555987, −2.63919403612049682151273130400, −2.36554791937411813960380152206, −1.36416374581958776184885824734, −0.58801804584055334542907052762,
0.58801804584055334542907052762, 1.36416374581958776184885824734, 2.36554791937411813960380152206, 2.63919403612049682151273130400, 3.27240944795662776286839555987, 3.40602800127778600023832834874, 4.45583216891991361716697981538, 4.74848220071396319869391504358, 5.15989707299736134545521197121, 5.45497924284220093302993069538, 5.76704191023427785652996088699, 6.06769170648307973716908360533, 6.65135054077798821651233858875, 7.10957752643818980496318820883, 7.65484550460326826376738680975, 7.79211370248316861524361187863, 8.076358446923671498948303426255, 8.440198218013408817418866236911, 9.163812027834394774548878849740, 9.315443038155154327358345653219
