| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 5·9-s − 4·14-s + 16-s − 6·17-s + 5·18-s + 12·23-s + 4·28-s + 4·31-s − 32-s + 6·34-s − 5·36-s − 6·41-s − 12·46-s + 24·47-s − 2·49-s − 4·56-s − 4·62-s − 20·63-s + 64-s − 6·68-s + 24·71-s + 5·72-s + 22·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 5/3·9-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 1.17·18-s + 2.50·23-s + 0.755·28-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 5/6·36-s − 0.937·41-s − 1.76·46-s + 3.50·47-s − 2/7·49-s − 0.534·56-s − 0.508·62-s − 2.51·63-s + 1/8·64-s − 0.727·68-s + 2.84·71-s + 0.589·72-s + 2.57·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.078912628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078912628\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.447991916742081591691197868932, −9.108795455289480583011416986557, −8.746535765801052966997753191886, −8.316812877075225116243141130520, −7.945377299913127727532994829665, −7.29853134793764088505009214313, −6.69992826322664140620538681665, −6.29316757927509488643773547834, −5.29439943415338779506744522168, −5.20592148788576143134616479765, −4.48531193864795315694041050402, −3.53176922528875095704922238510, −2.65155040417235216240951299295, −2.18562128408643501857872580519, −0.919526999450248053192487619525,
0.919526999450248053192487619525, 2.18562128408643501857872580519, 2.65155040417235216240951299295, 3.53176922528875095704922238510, 4.48531193864795315694041050402, 5.20592148788576143134616479765, 5.29439943415338779506744522168, 6.29316757927509488643773547834, 6.69992826322664140620538681665, 7.29853134793764088505009214313, 7.945377299913127727532994829665, 8.316812877075225116243141130520, 8.746535765801052966997753191886, 9.108795455289480583011416986557, 9.447991916742081591691197868932
