| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 4·13-s + 5·16-s − 4·19-s − 6·23-s − 10·25-s + 8·26-s − 10·31-s − 6·32-s − 8·37-s + 8·38-s + 6·41-s + 4·43-s + 12·46-s + 18·47-s + 20·50-s − 12·52-s − 4·61-s + 20·62-s + 7·64-s − 8·67-s + 6·71-s + 14·73-s + 16·74-s − 12·76-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 1.10·13-s + 5/4·16-s − 0.917·19-s − 1.25·23-s − 2·25-s + 1.56·26-s − 1.79·31-s − 1.06·32-s − 1.31·37-s + 1.29·38-s + 0.937·41-s + 0.609·43-s + 1.76·46-s + 2.62·47-s + 2.82·50-s − 1.66·52-s − 0.512·61-s + 2.54·62-s + 7/8·64-s − 0.977·67-s + 0.712·71-s + 1.63·73-s + 1.85·74-s − 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5613636064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5613636064\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 157 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 108 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 133 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 292 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.87070715053170823665148824684, −7.58450061834049440058570939850, −7.48209480296337241956066241069, −7.37544337930295583241073886394, −6.56350965543370507049476047150, −6.47065448065658464688099859166, −5.93757854295158494591442772512, −5.85916965054398795580096423075, −5.22960646183362394331565593093, −5.05454375529657728504916044836, −4.39883011103074252443831994009, −3.90771946877330373734870805123, −3.73546583493542228223149707050, −3.34523234861580284639266715671, −2.46375194319894047207926110135, −2.28370946523043075161120171921, −2.04968299535176241420589042651, −1.61775910309022841384645169948, −0.73843678087101198219298018542, −0.29759316981142721311044145931,
0.29759316981142721311044145931, 0.73843678087101198219298018542, 1.61775910309022841384645169948, 2.04968299535176241420589042651, 2.28370946523043075161120171921, 2.46375194319894047207926110135, 3.34523234861580284639266715671, 3.73546583493542228223149707050, 3.90771946877330373734870805123, 4.39883011103074252443831994009, 5.05454375529657728504916044836, 5.22960646183362394331565593093, 5.85916965054398795580096423075, 5.93757854295158494591442772512, 6.47065448065658464688099859166, 6.56350965543370507049476047150, 7.37544337930295583241073886394, 7.48209480296337241956066241069, 7.58450061834049440058570939850, 7.87070715053170823665148824684
