| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 3·5-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s + 6·10-s + 7·11-s − 6·12-s + 4·14-s + 6·15-s + 5·16-s + 17-s − 6·18-s − 19-s − 9·20-s + 4·21-s − 14·22-s + 23-s + 8·24-s + 25-s − 4·27-s − 6·28-s − 29-s − 12·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.34·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.89·10-s + 2.11·11-s − 1.73·12-s + 1.06·14-s + 1.54·15-s + 5/4·16-s + 0.242·17-s − 1.41·18-s − 0.229·19-s − 2.01·20-s + 0.872·21-s − 2.98·22-s + 0.208·23-s + 1.63·24-s + 1/5·25-s − 0.769·27-s − 1.13·28-s − 0.185·29-s − 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6722407694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6722407694\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 17 T + 154 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 118 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 19 T + 232 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.950582076466940698145219677834, −7.87473873718952103891309868393, −7.24833967352261023685898319759, −7.21940195867006375357813191464, −6.73188132240621946240011159912, −6.54970724765599673234416140709, −6.08346639879335781299519797261, −5.98809613404291535315695424318, −5.36267124498753473992275493470, −5.07760720067687761371554941761, −4.22998135057417449139168524993, −4.19617924627748042340852354755, −3.72381044853658398152580713167, −3.56513794513188689145523150626, −2.89827504331595590276262707192, −2.31638597677632616093771633069, −1.82597683556651211561604258599, −1.18989435134740020733538435282, −0.75371095167969600174767616774, −0.44180928026596389021966686337,
0.44180928026596389021966686337, 0.75371095167969600174767616774, 1.18989435134740020733538435282, 1.82597683556651211561604258599, 2.31638597677632616093771633069, 2.89827504331595590276262707192, 3.56513794513188689145523150626, 3.72381044853658398152580713167, 4.19617924627748042340852354755, 4.22998135057417449139168524993, 5.07760720067687761371554941761, 5.36267124498753473992275493470, 5.98809613404291535315695424318, 6.08346639879335781299519797261, 6.54970724765599673234416140709, 6.73188132240621946240011159912, 7.21940195867006375357813191464, 7.24833967352261023685898319759, 7.87473873718952103891309868393, 7.950582076466940698145219677834
