| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 4·8-s + 3·9-s − 4·10-s − 4·11-s − 6·12-s − 2·13-s − 4·15-s + 5·16-s + 8·17-s − 6·18-s − 10·19-s + 6·20-s + 8·22-s − 10·23-s + 8·24-s + 4·26-s − 4·27-s − 2·29-s + 8·30-s + 4·31-s − 6·32-s + 8·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 1.41·8-s + 9-s − 1.26·10-s − 1.20·11-s − 1.73·12-s − 0.554·13-s − 1.03·15-s + 5/4·16-s + 1.94·17-s − 1.41·18-s − 2.29·19-s + 1.34·20-s + 1.70·22-s − 2.08·23-s + 1.63·24-s + 0.784·26-s − 0.769·27-s − 0.371·29-s + 1.46·30-s + 0.718·31-s − 1.06·32-s + 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5193030716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5193030716\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 19 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 43 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 64 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 31 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 100 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 75 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 22 T + 235 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 22 T + 260 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 180 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.510113237566213736411873754387, −8.167665830533896014241873483105, −7.900086815364587875817381732845, −7.88104163522829595774433616819, −7.28614756517672543503092414402, −6.78139031334244418283531266880, −6.38694771981900348351044676853, −6.21401692202052580385421712943, −5.73867363088135577684018363039, −5.59766974614110183227624729728, −5.09070356934360039320653836404, −4.67129913836257447377357796544, −3.96408067604631106084150176420, −3.82222687670851610781373462422, −2.78967401544077329911184648715, −2.59368521918941962983976828447, −1.88528931128501600300237723586, −1.83280741425929237456236240185, −0.871994862396183004263668419787, −0.35754301108732754539049450301,
0.35754301108732754539049450301, 0.871994862396183004263668419787, 1.83280741425929237456236240185, 1.88528931128501600300237723586, 2.59368521918941962983976828447, 2.78967401544077329911184648715, 3.82222687670851610781373462422, 3.96408067604631106084150176420, 4.67129913836257447377357796544, 5.09070356934360039320653836404, 5.59766974614110183227624729728, 5.73867363088135577684018363039, 6.21401692202052580385421712943, 6.38694771981900348351044676853, 6.78139031334244418283531266880, 7.28614756517672543503092414402, 7.88104163522829595774433616819, 7.900086815364587875817381732845, 8.167665830533896014241873483105, 8.510113237566213736411873754387
