| L(s) = 1 | + 2·5-s − 2·7-s + 8·13-s + 10·17-s + 2·19-s − 8·23-s − 4·25-s − 2·29-s − 4·35-s − 8·37-s + 16·41-s − 4·43-s − 6·47-s + 3·49-s + 6·53-s + 16·59-s + 16·61-s + 16·65-s − 8·67-s − 2·71-s + 8·73-s − 4·79-s + 10·83-s + 20·85-s − 16·91-s + 4·95-s + 12·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 2.21·13-s + 2.42·17-s + 0.458·19-s − 1.66·23-s − 4/5·25-s − 0.371·29-s − 0.676·35-s − 1.31·37-s + 2.49·41-s − 0.609·43-s − 0.875·47-s + 3/7·49-s + 0.824·53-s + 2.08·59-s + 2.04·61-s + 1.98·65-s − 0.977·67-s − 0.237·71-s + 0.936·73-s − 0.450·79-s + 1.09·83-s + 2.16·85-s − 1.67·91-s + 0.410·95-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.853021497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.853021497\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 56 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 116 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 164 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−7.82508858757232868766556216147, −7.48966471687452736005384545564, −7.30300186993056739819950634438, −6.72033452185198052128553924444, −6.28104305201082942431105533896, −6.16354047907991182134082748996, −5.78211797709902953350540770050, −5.69475418493493397485105148642, −5.15437858554929948956192632811, −5.06474179561794531607418513894, −3.97313550308581969774860911952, −3.96967273762296694840005577259, −3.55399166021064744249868098288, −3.55163173685440234926923557660, −2.68504311789875791285292515414, −2.53939371161969454489779208379, −1.71977579391393835244646285291, −1.62569836511552949477421568566, −0.950352055993476671989546866435, −0.57458679088412248307946885761,
0.57458679088412248307946885761, 0.950352055993476671989546866435, 1.62569836511552949477421568566, 1.71977579391393835244646285291, 2.53939371161969454489779208379, 2.68504311789875791285292515414, 3.55163173685440234926923557660, 3.55399166021064744249868098288, 3.96967273762296694840005577259, 3.97313550308581969774860911952, 5.06474179561794531607418513894, 5.15437858554929948956192632811, 5.69475418493493397485105148642, 5.78211797709902953350540770050, 6.16354047907991182134082748996, 6.28104305201082942431105533896, 6.72033452185198052128553924444, 7.30300186993056739819950634438, 7.48966471687452736005384545564, 7.82508858757232868766556216147
