| L(s) = 1 | − 2·3-s + 6·5-s + 3·9-s − 12·15-s + 6·17-s + 8·19-s + 17·25-s − 4·27-s − 8·31-s + 18·45-s + 11·49-s − 12·51-s − 16·57-s + 24·59-s + 14·61-s + 16·67-s + 24·71-s − 10·73-s − 34·75-s − 16·79-s + 5·81-s + 36·85-s + 16·93-s + 48·95-s + 12·101-s − 16·103-s − 5·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2.68·5-s + 9-s − 3.09·15-s + 1.45·17-s + 1.83·19-s + 17/5·25-s − 0.769·27-s − 1.43·31-s + 2.68·45-s + 11/7·49-s − 1.68·51-s − 2.11·57-s + 3.12·59-s + 1.79·61-s + 1.95·67-s + 2.84·71-s − 1.17·73-s − 3.92·75-s − 1.80·79-s + 5/9·81-s + 3.90·85-s + 1.65·93-s + 4.92·95-s + 1.19·101-s − 1.57·103-s − 0.454·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.921070907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.921070907\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−10.11212990148886276707591589751, −10.02017537361485986293543166648, −9.684591119540926845075330593299, −9.270551215971529240756058463732, −8.857587481490713924203218630949, −8.261569506078027520851955170996, −7.45908154591471216472748035463, −7.35510016162596689830124967140, −6.60330487448318824043062551817, −6.44838742050001858017331612020, −5.75012242312882338377866210208, −5.42497060281067294429038301638, −5.27224593357481154922089293088, −5.17507351979264049571677262892, −3.81942794175715741952769088228, −3.69655934440246266367533170301, −2.48408618451242772424447905101, −2.32804339716000629728928095014, −1.31592931519838858223283844811, −1.05373904367424453501139955617,
1.05373904367424453501139955617, 1.31592931519838858223283844811, 2.32804339716000629728928095014, 2.48408618451242772424447905101, 3.69655934440246266367533170301, 3.81942794175715741952769088228, 5.17507351979264049571677262892, 5.27224593357481154922089293088, 5.42497060281067294429038301638, 5.75012242312882338377866210208, 6.44838742050001858017331612020, 6.60330487448318824043062551817, 7.35510016162596689830124967140, 7.45908154591471216472748035463, 8.261569506078027520851955170996, 8.857587481490713924203218630949, 9.270551215971529240756058463732, 9.684591119540926845075330593299, 10.02017537361485986293543166648, 10.11212990148886276707591589751
