| L(s) = 1 | − 2·2-s − 2·3-s − 4-s − 4·5-s + 4·6-s + 8·8-s + 3·9-s + 8·10-s − 4·11-s + 2·12-s − 4·13-s + 8·15-s − 7·16-s − 6·18-s − 4·19-s + 4·20-s + 8·22-s − 2·23-s − 16·24-s + 2·25-s + 8·26-s − 4·27-s + 2·29-s − 16·30-s − 16·31-s − 14·32-s + 8·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s − 1/2·4-s − 1.78·5-s + 1.63·6-s + 2.82·8-s + 9-s + 2.52·10-s − 1.20·11-s + 0.577·12-s − 1.10·13-s + 2.06·15-s − 7/4·16-s − 1.41·18-s − 0.917·19-s + 0.894·20-s + 1.70·22-s − 0.417·23-s − 3.26·24-s + 2/5·25-s + 1.56·26-s − 0.769·27-s + 0.371·29-s − 2.92·30-s − 2.87·31-s − 2.47·32-s + 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07670287366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07670287366\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.338447511055055077126074365178, −9.097503522664397494269737288047, −8.362387586883570179645531607502, −8.250225379137463814134942452041, −7.69887750453685184898414395008, −7.66838139848996064386030734580, −7.23068243215044427121692976563, −7.04343517956396977846239193971, −6.08054193823677989389679726765, −5.84583734030000226310010687063, −5.08750031272530907332371021223, −4.94425588189532455254938780655, −4.40136335005202697510961132096, −4.34074582951104387990750663525, −3.51216822143666048679181254394, −3.37056710671189899513123215829, −1.97248292149484179888563619549, −1.84492219283261523096541628336, −0.51485693571527491062728122169, −0.30344002975376199490367420995,
0.30344002975376199490367420995, 0.51485693571527491062728122169, 1.84492219283261523096541628336, 1.97248292149484179888563619549, 3.37056710671189899513123215829, 3.51216822143666048679181254394, 4.34074582951104387990750663525, 4.40136335005202697510961132096, 4.94425588189532455254938780655, 5.08750031272530907332371021223, 5.84583734030000226310010687063, 6.08054193823677989389679726765, 7.04343517956396977846239193971, 7.23068243215044427121692976563, 7.66838139848996064386030734580, 7.69887750453685184898414395008, 8.250225379137463814134942452041, 8.362387586883570179645531607502, 9.097503522664397494269737288047, 9.338447511055055077126074365178
