| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s + 2·7-s − 4·8-s + 4·10-s + 4·13-s − 4·14-s + 5·16-s − 2·17-s − 4·19-s − 6·20-s − 2·23-s − 4·25-s − 8·26-s + 6·28-s − 16·29-s − 10·31-s − 6·32-s + 4·34-s − 4·35-s + 8·37-s + 8·38-s + 8·40-s + 4·41-s + 4·46-s + 6·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s − 1.41·8-s + 1.26·10-s + 1.10·13-s − 1.06·14-s + 5/4·16-s − 0.485·17-s − 0.917·19-s − 1.34·20-s − 0.417·23-s − 4/5·25-s − 1.56·26-s + 1.13·28-s − 2.97·29-s − 1.79·31-s − 1.06·32-s + 0.685·34-s − 0.676·35-s + 1.31·37-s + 1.29·38-s + 1.26·40-s + 0.624·41-s + 0.589·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8398404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 104 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 74 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 232 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 240 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.419844399867227504734333844765, −8.418385845216893773065805396349, −7.76153100644054777353324239760, −7.61366718325074957265050904203, −7.21122372819985152531563685800, −7.07562304732909573071995871292, −6.30020894599441998167433158157, −5.92903874395923770611690307388, −5.69036877399604018393798690692, −5.34096492615294192951695102539, −4.39497980279158690824289828256, −4.17762520132954813782859672028, −3.73662718221566422543883226726, −3.49535244114461844856153563601, −2.47046372628679781854104138469, −2.30385092076484685652096088854, −1.54777570597275483446057107967, −1.30279597213553672093675431844, 0, 0,
1.30279597213553672093675431844, 1.54777570597275483446057107967, 2.30385092076484685652096088854, 2.47046372628679781854104138469, 3.49535244114461844856153563601, 3.73662718221566422543883226726, 4.17762520132954813782859672028, 4.39497980279158690824289828256, 5.34096492615294192951695102539, 5.69036877399604018393798690692, 5.92903874395923770611690307388, 6.30020894599441998167433158157, 7.07562304732909573071995871292, 7.21122372819985152531563685800, 7.61366718325074957265050904203, 7.76153100644054777353324239760, 8.418385845216893773065805396349, 8.419844399867227504734333844765
