| L(s) = 1 | + 2·5-s + 3·7-s − 2·11-s + 2·13-s − 7·17-s + 9·19-s − 6·23-s + 3·25-s + 5·29-s + 5·31-s + 6·35-s − 37-s + 20·41-s + 2·43-s + 6·47-s − 3·49-s + 13·53-s − 4·55-s + 2·59-s − 3·61-s + 4·65-s + 9·71-s + 18·73-s − 6·77-s + 10·79-s − 20·83-s − 14·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.13·7-s − 0.603·11-s + 0.554·13-s − 1.69·17-s + 2.06·19-s − 1.25·23-s + 3/5·25-s + 0.928·29-s + 0.898·31-s + 1.01·35-s − 0.164·37-s + 3.12·41-s + 0.304·43-s + 0.875·47-s − 3/7·49-s + 1.78·53-s − 0.539·55-s + 0.260·59-s − 0.384·61-s + 0.496·65-s + 1.06·71-s + 2.10·73-s − 0.683·77-s + 1.12·79-s − 2.19·83-s − 1.51·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.074674934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.074674934\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 54 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 120 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 158 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 26 T + 346 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
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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
−8.538635385706729659685077755658, −8.320278943682862826452854972727, −7.82429152551889613050976037380, −7.74282792418133725170019263166, −7.10616330976738281304077416408, −6.92801450965601876153003652734, −6.23338359420560625302644137487, −6.13569029005216695240034415597, −5.51667357525040713261715151201, −5.48716972949232146085652147968, −4.81601499484471995067949897508, −4.56197546287805922151796010638, −4.21112239957441019596349478859, −3.67880596935055625036903281520, −3.07887933862199104193716312538, −2.56445394667358042603938767457, −2.18178746378526174617723301059, −1.91391709410738428672465169812, −0.889077701804226795350455029179, −0.881744750176254648638780428010,
0.881744750176254648638780428010, 0.889077701804226795350455029179, 1.91391709410738428672465169812, 2.18178746378526174617723301059, 2.56445394667358042603938767457, 3.07887933862199104193716312538, 3.67880596935055625036903281520, 4.21112239957441019596349478859, 4.56197546287805922151796010638, 4.81601499484471995067949897508, 5.48716972949232146085652147968, 5.51667357525040713261715151201, 6.13569029005216695240034415597, 6.23338359420560625302644137487, 6.92801450965601876153003652734, 7.10616330976738281304077416408, 7.74282792418133725170019263166, 7.82429152551889613050976037380, 8.320278943682862826452854972727, 8.538635385706729659685077755658
