| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s + 4·7-s − 4·8-s + 4·10-s + 2·11-s − 8·14-s + 5·16-s + 4·17-s + 6·19-s − 6·20-s − 4·22-s + 2·23-s + 12·28-s + 8·29-s + 8·31-s − 6·32-s − 8·34-s − 8·35-s − 2·37-s − 12·38-s + 8·40-s + 12·41-s + 6·43-s + 6·44-s − 4·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.51·7-s − 1.41·8-s + 1.26·10-s + 0.603·11-s − 2.13·14-s + 5/4·16-s + 0.970·17-s + 1.37·19-s − 1.34·20-s − 0.852·22-s + 0.417·23-s + 2.26·28-s + 1.48·29-s + 1.43·31-s − 1.06·32-s − 1.37·34-s − 1.35·35-s − 0.328·37-s − 1.94·38-s + 1.26·40-s + 1.87·41-s + 0.914·43-s + 0.904·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.060427061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060427061\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 100 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 120 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 122 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 280 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 182 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−11.42040563994266866501743132035, −10.98155484782329337888949113709, −10.40482599114974020735257880798, −10.26095027768618246511897003410, −9.378722773826353719771765495979, −9.312501618093317564853641216025, −8.542776875790259506043258999385, −8.287703697821199902486276496520, −7.74513055401376614517540185136, −7.59310279482573095032487397250, −7.07016137321053242716383940365, −6.48938586343200272739556563697, −5.69890845920604040060471151389, −5.37637725804369239791115941708, −4.36398200018279965241270343877, −4.22026374078375503376121968005, −3.05176169464901095133350979583, −2.67311810030499191184477268671, −1.36365755960770692493242196130, −1.03252784750203758561527897719,
1.03252784750203758561527897719, 1.36365755960770692493242196130, 2.67311810030499191184477268671, 3.05176169464901095133350979583, 4.22026374078375503376121968005, 4.36398200018279965241270343877, 5.37637725804369239791115941708, 5.69890845920604040060471151389, 6.48938586343200272739556563697, 7.07016137321053242716383940365, 7.59310279482573095032487397250, 7.74513055401376614517540185136, 8.287703697821199902486276496520, 8.542776875790259506043258999385, 9.312501618093317564853641216025, 9.378722773826353719771765495979, 10.26095027768618246511897003410, 10.40482599114974020735257880798, 10.98155484782329337888949113709, 11.42040563994266866501743132035
