| L(s) = 1 | − 2·5-s − 2·7-s − 11-s + 13-s − 11·17-s + 6·19-s − 2·23-s + 3·25-s − 5·29-s + 4·31-s + 4·35-s − 6·41-s + 6·43-s + 9·47-s + 3·49-s + 18·53-s + 2·55-s − 8·59-s − 22·61-s − 2·65-s + 12·67-s + 8·73-s + 2·77-s + 11·79-s + 24·83-s + 22·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 0.301·11-s + 0.277·13-s − 2.66·17-s + 1.37·19-s − 0.417·23-s + 3/5·25-s − 0.928·29-s + 0.718·31-s + 0.676·35-s − 0.937·41-s + 0.914·43-s + 1.31·47-s + 3/7·49-s + 2.47·53-s + 0.269·55-s − 1.04·59-s − 2.81·61-s − 0.248·65-s + 1.46·67-s + 0.936·73-s + 0.227·77-s + 1.23·79-s + 2.63·83-s + 2.38·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.520203770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520203770\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 22 T + 226 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11 T + 150 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 212 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−8.376352933923987697881462652721, −7.943953711985302313599634157599, −7.75129804567370188099723043578, −7.35097072386167772473578576793, −6.90142415113962653825254449128, −6.77754797808308725645122249598, −6.19893617169070823099912114409, −6.09118347919350747018662874290, −5.41880029142938382635313992103, −5.16693274618314918366236839141, −4.59862911169441425755650110230, −4.37329107919533509929174439718, −3.78674735455878598857505344542, −3.71526835906414727112246254830, −3.06128028583995456087631014047, −2.70367589166985038959217744102, −2.16597998009413846798849039409, −1.80329695508258457010194902872, −0.75485493081111532783414696023, −0.46766618822769760800535659233,
0.46766618822769760800535659233, 0.75485493081111532783414696023, 1.80329695508258457010194902872, 2.16597998009413846798849039409, 2.70367589166985038959217744102, 3.06128028583995456087631014047, 3.71526835906414727112246254830, 3.78674735455878598857505344542, 4.37329107919533509929174439718, 4.59862911169441425755650110230, 5.16693274618314918366236839141, 5.41880029142938382635313992103, 6.09118347919350747018662874290, 6.19893617169070823099912114409, 6.77754797808308725645122249598, 6.90142415113962653825254449128, 7.35097072386167772473578576793, 7.75129804567370188099723043578, 7.943953711985302313599634157599, 8.376352933923987697881462652721
