| L(s) = 1 | + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 4·13-s − 8·14-s + 5·16-s − 2·19-s − 6·23-s − 8·26-s − 12·28-s + 4·31-s + 6·32-s − 16·37-s − 4·38-s − 4·43-s − 12·46-s − 6·47-s + 49-s − 12·52-s − 18·53-s − 16·56-s + 4·61-s + 8·62-s + 7·64-s − 16·67-s + 12·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 1.10·13-s − 2.13·14-s + 5/4·16-s − 0.458·19-s − 1.25·23-s − 1.56·26-s − 2.26·28-s + 0.718·31-s + 1.06·32-s − 2.63·37-s − 0.648·38-s − 0.609·43-s − 1.76·46-s − 0.875·47-s + 1/7·49-s − 1.66·52-s − 2.47·53-s − 2.13·56-s + 0.512·61-s + 1.01·62-s + 7/8·64-s − 1.95·67-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T - 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 175 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.016948402030064565863561924130, −7.930634814744980147377337556239, −7.28316109088332046334786287334, −6.99697371067983458645161584759, −6.58212775219323659447922185535, −6.49082823515108118429248150928, −6.08115872466775017042183637807, −5.56386009620794927755614690211, −5.26347936668625295831886516238, −4.92071088805352290561763617906, −4.29280623860668206962206143488, −4.23010252821686291171883985852, −3.47343377950167578428961548105, −3.34933732230151683333544520629, −2.78919757863597125203275196827, −2.58180466540315450577668951260, −1.69868254046163333588646506862, −1.59137209604272726220549827532, 0, 0,
1.59137209604272726220549827532, 1.69868254046163333588646506862, 2.58180466540315450577668951260, 2.78919757863597125203275196827, 3.34933732230151683333544520629, 3.47343377950167578428961548105, 4.23010252821686291171883985852, 4.29280623860668206962206143488, 4.92071088805352290561763617906, 5.26347936668625295831886516238, 5.56386009620794927755614690211, 6.08115872466775017042183637807, 6.49082823515108118429248150928, 6.58212775219323659447922185535, 6.99697371067983458645161584759, 7.28316109088332046334786287334, 7.930634814744980147377337556239, 8.016948402030064565863561924130
