| L(s) = 1 | − 2·2-s + 4-s + 2·5-s − 4·10-s − 4·11-s + 4·13-s + 16-s + 4·17-s + 2·20-s + 8·22-s + 2·23-s + 3·25-s − 8·26-s + 2·29-s + 12·31-s + 2·32-s − 8·34-s − 10·41-s + 10·43-s − 4·44-s − 4·46-s + 4·47-s − 6·50-s + 4·52-s + 8·53-s − 8·55-s − 4·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.894·5-s − 1.26·10-s − 1.20·11-s + 1.10·13-s + 1/4·16-s + 0.970·17-s + 0.447·20-s + 1.70·22-s + 0.417·23-s + 3/5·25-s − 1.56·26-s + 0.371·29-s + 2.15·31-s + 0.353·32-s − 1.37·34-s − 1.56·41-s + 1.52·43-s − 0.603·44-s − 0.589·46-s + 0.583·47-s − 0.848·50-s + 0.554·52-s + 1.09·53-s − 1.07·55-s − 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.400145556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400145556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 22 T + 253 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.081903751884427821723928215700, −9.062140330744835558620378449898, −8.316015890782907821499227805975, −8.295051618054720759783093170487, −7.903131127127360347513550397427, −7.64780582155133171512515723040, −6.77696520387078489263108014111, −6.63073814638768040818137817348, −6.32196212560656542401888182228, −5.59433109814177910829958867179, −5.36843100655238943246494750032, −5.14708635134894939708903978830, −4.34752971437381522600465814837, −3.98531378766869734644400305545, −3.21631757366628341768671418290, −2.85878372252449078851070924427, −2.39356275303105450313270662956, −1.72265950427876969138154289759, −0.879607370469596228051236949514, −0.75042265159337449102848382756,
0.75042265159337449102848382756, 0.879607370469596228051236949514, 1.72265950427876969138154289759, 2.39356275303105450313270662956, 2.85878372252449078851070924427, 3.21631757366628341768671418290, 3.98531378766869734644400305545, 4.34752971437381522600465814837, 5.14708635134894939708903978830, 5.36843100655238943246494750032, 5.59433109814177910829958867179, 6.32196212560656542401888182228, 6.63073814638768040818137817348, 6.77696520387078489263108014111, 7.64780582155133171512515723040, 7.903131127127360347513550397427, 8.295051618054720759783093170487, 8.316015890782907821499227805975, 9.062140330744835558620378449898, 9.081903751884427821723928215700
