| L(s) = 1 | − 2·2-s + 3·4-s + 4·5-s − 4·8-s − 8·10-s + 2·11-s − 8·13-s + 5·16-s + 4·17-s − 4·19-s + 12·20-s − 4·22-s + 4·23-s + 2·25-s + 16·26-s − 8·29-s − 4·31-s − 6·32-s − 8·34-s − 4·37-s + 8·38-s − 16·40-s + 12·41-s − 4·43-s + 6·44-s − 8·46-s + 4·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.78·5-s − 1.41·8-s − 2.52·10-s + 0.603·11-s − 2.21·13-s + 5/4·16-s + 0.970·17-s − 0.917·19-s + 2.68·20-s − 0.852·22-s + 0.834·23-s + 2/5·25-s + 3.13·26-s − 1.48·29-s − 0.718·31-s − 1.06·32-s − 1.37·34-s − 0.657·37-s + 1.29·38-s − 2.52·40-s + 1.87·41-s − 0.609·43-s + 0.904·44-s − 1.17·46-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 136 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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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
−7.43077296037992944101561196873, −7.25400396987864757599306270862, −6.99381800976401536451564333563, −6.61169619931255753045439163928, −6.14474661702482663474943246708, −5.79072982294630856807743882864, −5.59135440786651729989460371137, −5.45066164102003787373877681616, −4.69426683598062010999446331213, −4.61370304557787753222486385129, −3.76369975183680166369929419344, −3.66631857556829425311793961539, −2.77129485627538823884269245893, −2.69109314124282389346293847004, −2.10715122356964009386802956626, −2.06395870865954971515008918928, −1.27413014813717384512932744551, −1.26760130220323677632751335347, 0, 0,
1.26760130220323677632751335347, 1.27413014813717384512932744551, 2.06395870865954971515008918928, 2.10715122356964009386802956626, 2.69109314124282389346293847004, 2.77129485627538823884269245893, 3.66631857556829425311793961539, 3.76369975183680166369929419344, 4.61370304557787753222486385129, 4.69426683598062010999446331213, 5.45066164102003787373877681616, 5.59135440786651729989460371137, 5.79072982294630856807743882864, 6.14474661702482663474943246708, 6.61169619931255753045439163928, 6.99381800976401536451564333563, 7.25400396987864757599306270862, 7.43077296037992944101561196873
