| L(s) = 1 | + 2·2-s + 3-s + 3·4-s − 2·5-s + 2·6-s + 2·7-s + 4·8-s − 4·9-s − 4·10-s + 2·11-s + 3·12-s + 5·13-s + 4·14-s − 2·15-s + 5·16-s + 6·17-s − 8·18-s − 3·19-s − 6·20-s + 2·21-s + 4·22-s + 4·23-s + 4·24-s − 2·25-s + 10·26-s − 6·27-s + 6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.894·5-s + 0.816·6-s + 0.755·7-s + 1.41·8-s − 4/3·9-s − 1.26·10-s + 0.603·11-s + 0.866·12-s + 1.38·13-s + 1.06·14-s − 0.516·15-s + 5/4·16-s + 1.45·17-s − 1.88·18-s − 0.688·19-s − 1.34·20-s + 0.436·21-s + 0.852·22-s + 0.834·23-s + 0.816·24-s − 2/5·25-s + 1.96·26-s − 1.15·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2172676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2172676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.991389727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.991389727\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 67 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 21 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 81 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 83 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 130 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 22 T + 210 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T - 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 122 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 25 T + 333 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.410779865811157774209825111746, −9.308018786529995840725112393482, −8.776958444780189431030496196100, −8.524651487899296611288587108151, −7.85846809563325517492147536512, −7.70072067029257885292313364068, −7.36585319598698788417869973772, −6.99307453196890550547625290569, −6.01958517200536420362726030583, −5.92870284229866561399675564250, −5.53627531310541720006188457182, −5.40246969615894539996111836433, −4.35602153984041629118806922037, −4.03638720546347678634076831936, −3.82333143663412623733839984282, −3.46178655200750915808725869609, −2.73206898819433609853926364121, −2.38349148889661955832382093658, −1.61406431983922960611906333089, −0.836663454717212192414894026082,
0.836663454717212192414894026082, 1.61406431983922960611906333089, 2.38349148889661955832382093658, 2.73206898819433609853926364121, 3.46178655200750915808725869609, 3.82333143663412623733839984282, 4.03638720546347678634076831936, 4.35602153984041629118806922037, 5.40246969615894539996111836433, 5.53627531310541720006188457182, 5.92870284229866561399675564250, 6.01958517200536420362726030583, 6.99307453196890550547625290569, 7.36585319598698788417869973772, 7.70072067029257885292313364068, 7.85846809563325517492147536512, 8.524651487899296611288587108151, 8.776958444780189431030496196100, 9.308018786529995840725112393482, 9.410779865811157774209825111746
