| L(s) = 1 | + 2·5-s + 7-s + 5·11-s + 5·13-s + 4·17-s + 2·19-s − 7·23-s + 3·25-s + 5·29-s + 5·31-s + 2·35-s − 6·37-s + 8·43-s + 5·47-s + 49-s + 9·53-s + 10·55-s − 26·59-s + 2·61-s + 10·65-s − 14·67-s − 9·71-s + 5·77-s + 6·79-s + 10·83-s + 8·85-s + 3·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s + 1.50·11-s + 1.38·13-s + 0.970·17-s + 0.458·19-s − 1.45·23-s + 3/5·25-s + 0.928·29-s + 0.898·31-s + 0.338·35-s − 0.986·37-s + 1.21·43-s + 0.729·47-s + 1/7·49-s + 1.23·53-s + 1.34·55-s − 3.38·59-s + 0.256·61-s + 1.24·65-s − 1.71·67-s − 1.06·71-s + 0.569·77-s + 0.675·79-s + 1.09·83-s + 0.867·85-s + 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.356661242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.356661242\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 44 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 50 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 86 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 112 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 148 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 134 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−8.857449078695155521324121511884, −8.720762709195757263662235507376, −7.917225906548906442004130191206, −7.87589259388261715397818721514, −7.40472637544268798116812868346, −6.93137968145181140621046171731, −6.35870535421854895138558621178, −6.17766391193468089129121537353, −5.93421836585419308708142466618, −5.66778610808101394820397488795, −4.85053597454650241862841687672, −4.73688837916018369756851686823, −4.06095534746289918622858142520, −3.84430790532800964179544769989, −3.16432154335307708493304060967, −3.02314884188694069630644257720, −2.03295347094569248608226426374, −1.82764502667012712630972120616, −1.13902324336306079739609225599, −0.844460583937210432502758867826,
0.844460583937210432502758867826, 1.13902324336306079739609225599, 1.82764502667012712630972120616, 2.03295347094569248608226426374, 3.02314884188694069630644257720, 3.16432154335307708493304060967, 3.84430790532800964179544769989, 4.06095534746289918622858142520, 4.73688837916018369756851686823, 4.85053597454650241862841687672, 5.66778610808101394820397488795, 5.93421836585419308708142466618, 6.17766391193468089129121537353, 6.35870535421854895138558621178, 6.93137968145181140621046171731, 7.40472637544268798116812868346, 7.87589259388261715397818721514, 7.917225906548906442004130191206, 8.720762709195757263662235507376, 8.857449078695155521324121511884
