| L(s) = 1 | − 4·25-s + 16·37-s + 24·47-s + 14·49-s + 12·59-s − 48·83-s − 64·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4/5·25-s + 2.63·37-s + 3.50·47-s + 2·49-s + 1.56·59-s − 5.26·83-s − 6.13·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09836864239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09836864239\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.25814290978141918485124794394, −5.76735368766186348986066713867, −5.76515481745639191221129393818, −5.68676832819156409439262995130, −5.31975209763907443802360889622, −5.31055211203398636729193298299, −5.16247616849183284838502815948, −4.56799010705388635587468425687, −4.36417634175394116819730790853, −4.25992523959258297305155712626, −4.20501898254237431183322129056, −3.98886279730996688910234514508, −3.76370380319991488130551559939, −3.53012362507592307576924325945, −3.12030548979434185576113545120, −2.81114110682084141023608228566, −2.68029201200092424888316633124, −2.45556114013393721282496942550, −2.39462478219914622774970085267, −2.04106665427616184085299894784, −1.61894045361787117320079284327, −1.12833511570297928311796515283, −1.09622869285781823564288051050, −0.871220741973297946928450288144, −0.04664815682414418271997626973,
0.04664815682414418271997626973, 0.871220741973297946928450288144, 1.09622869285781823564288051050, 1.12833511570297928311796515283, 1.61894045361787117320079284327, 2.04106665427616184085299894784, 2.39462478219914622774970085267, 2.45556114013393721282496942550, 2.68029201200092424888316633124, 2.81114110682084141023608228566, 3.12030548979434185576113545120, 3.53012362507592307576924325945, 3.76370380319991488130551559939, 3.98886279730996688910234514508, 4.20501898254237431183322129056, 4.25992523959258297305155712626, 4.36417634175394116819730790853, 4.56799010705388635587468425687, 5.16247616849183284838502815948, 5.31055211203398636729193298299, 5.31975209763907443802360889622, 5.68676832819156409439262995130, 5.76515481745639191221129393818, 5.76735368766186348986066713867, 6.25814290978141918485124794394
