| L(s) = 1 | + 2·3-s − 2·7-s + 2·9-s − 4·11-s − 2·13-s − 4·17-s − 4·21-s − 8·23-s + 6·27-s + 10·29-s + 6·31-s − 8·33-s + 6·37-s − 4·39-s + 14·41-s − 8·43-s + 4·47-s + 3·49-s − 8·51-s + 8·53-s − 10·59-s − 6·61-s − 4·63-s + 4·67-s − 16·69-s − 4·71-s − 22·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 2/3·9-s − 1.20·11-s − 0.554·13-s − 0.970·17-s − 0.872·21-s − 1.66·23-s + 1.15·27-s + 1.85·29-s + 1.07·31-s − 1.39·33-s + 0.986·37-s − 0.640·39-s + 2.18·41-s − 1.21·43-s + 0.583·47-s + 3/7·49-s − 1.12·51-s + 1.09·53-s − 1.30·59-s − 0.768·61-s − 0.503·63-s + 0.488·67-s − 1.92·69-s − 0.474·71-s − 2.57·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.140712079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.140712079\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 133 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 101 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 22 T + 262 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 30 T + 398 T^{2} - 30 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 198 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.944092774874072713022000159236, −8.638528663210322841956875301491, −8.176594259748816723388482059003, −7.81305059374674176993384642630, −7.78653426732545113855173163768, −7.14672123513557077374370831644, −6.73049089923149401891755600668, −6.40791200957661863560432499613, −5.83185782036036797624653600596, −5.81055279427730804809688574258, −4.82537423588319404503854915512, −4.58734844747870512609635747189, −4.38940631618763918276018055355, −3.74757900485494694916709165600, −3.14712312545492318428225601076, −2.82988233079559492728731848036, −2.34640979880319563825822244964, −2.30125026175556499647069843523, −1.23852899901108490134879567304, −0.44260037962077578448885832730,
0.44260037962077578448885832730, 1.23852899901108490134879567304, 2.30125026175556499647069843523, 2.34640979880319563825822244964, 2.82988233079559492728731848036, 3.14712312545492318428225601076, 3.74757900485494694916709165600, 4.38940631618763918276018055355, 4.58734844747870512609635747189, 4.82537423588319404503854915512, 5.81055279427730804809688574258, 5.83185782036036797624653600596, 6.40791200957661863560432499613, 6.73049089923149401891755600668, 7.14672123513557077374370831644, 7.78653426732545113855173163768, 7.81305059374674176993384642630, 8.176594259748816723388482059003, 8.638528663210322841956875301491, 8.944092774874072713022000159236
