| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s + 4·10-s + 2·11-s − 6·12-s + 4·13-s − 4·14-s − 4·15-s + 5·16-s + 6·17-s + 6·18-s + 8·19-s + 6·20-s + 4·21-s + 4·22-s + 2·23-s − 8·24-s + 3·25-s + 8·26-s − 4·27-s − 6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s + 1.26·10-s + 0.603·11-s − 1.73·12-s + 1.10·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 1.45·17-s + 1.41·18-s + 1.83·19-s + 1.34·20-s + 0.872·21-s + 0.852·22-s + 0.417·23-s − 1.63·24-s + 3/5·25-s + 1.56·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.617676150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.617676150\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−10.70347967111563947986137354215, −10.38715393270090526032149908101, −9.893093454040967770924047872289, −9.727850804397065485484578916840, −9.079674642465288933737635760686, −8.600193152708372041192707561691, −7.72790984760758369525384065297, −7.47348668314169995430620001292, −6.80283966174642378824177880289, −6.46063291197120849703060187639, −6.19114578083713139793062404786, −5.66120849697089272939956247000, −5.18447987062167723771357045743, −5.16330293180140880866797513771, −4.21035356013394717146926553200, −3.70711499408325527712209289758, −3.19008979336382757358983849240, −2.73794016074583526324836268727, −1.41641795769549472108387942963, −1.22689005674514361395070258668,
1.22689005674514361395070258668, 1.41641795769549472108387942963, 2.73794016074583526324836268727, 3.19008979336382757358983849240, 3.70711499408325527712209289758, 4.21035356013394717146926553200, 5.16330293180140880866797513771, 5.18447987062167723771357045743, 5.66120849697089272939956247000, 6.19114578083713139793062404786, 6.46063291197120849703060187639, 6.80283966174642378824177880289, 7.47348668314169995430620001292, 7.72790984760758369525384065297, 8.600193152708372041192707561691, 9.079674642465288933737635760686, 9.727850804397065485484578916840, 9.893093454040967770924047872289, 10.38715393270090526032149908101, 10.70347967111563947986137354215
