| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s − 4·8-s + 4·10-s − 4·11-s + 4·13-s + 5·16-s − 4·17-s − 2·19-s − 6·20-s + 8·22-s − 6·23-s + 3·25-s − 8·26-s − 4·31-s − 6·32-s + 8·34-s + 4·37-s + 4·38-s + 8·40-s + 8·41-s + 12·43-s − 12·44-s + 12·46-s + 2·47-s − 6·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.41·8-s + 1.26·10-s − 1.20·11-s + 1.10·13-s + 5/4·16-s − 0.970·17-s − 0.458·19-s − 1.34·20-s + 1.70·22-s − 1.25·23-s + 3/5·25-s − 1.56·26-s − 0.718·31-s − 1.06·32-s + 1.37·34-s + 0.657·37-s + 0.648·38-s + 1.26·40-s + 1.24·41-s + 1.82·43-s − 1.80·44-s + 1.76·46-s + 0.291·47-s − 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9014461692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9014461692\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 + 4 T + 19 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 91 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T - 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 103 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 20 T + 206 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 238 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−8.297758750482799352906926011900, −8.136849182982014457976106887619, −7.931905328461826550762469704699, −7.74410788786015766715830926302, −7.26109674022931713265655303244, −6.78633346794769802126916590369, −6.44460242093115122633522163736, −6.21018311684457643895745762607, −5.64157555586781990281115174484, −5.44553828256478317633364518976, −4.75945328574251872505089478810, −4.25867276082012604872874599571, −4.02193877972664700730921402366, −3.54974935081577372400791081908, −2.88301552095460879657484639749, −2.68678158768468436125366731685, −1.88119238682912980248616052906, −1.85169748182741527303104406663, −0.61920539799761890909604882660, −0.57567148317144328754404642758,
0.57567148317144328754404642758, 0.61920539799761890909604882660, 1.85169748182741527303104406663, 1.88119238682912980248616052906, 2.68678158768468436125366731685, 2.88301552095460879657484639749, 3.54974935081577372400791081908, 4.02193877972664700730921402366, 4.25867276082012604872874599571, 4.75945328574251872505089478810, 5.44553828256478317633364518976, 5.64157555586781990281115174484, 6.21018311684457643895745762607, 6.44460242093115122633522163736, 6.78633346794769802126916590369, 7.26109674022931713265655303244, 7.74410788786015766715830926302, 7.931905328461826550762469704699, 8.136849182982014457976106887619, 8.297758750482799352906926011900
