| L(s) = 1 | + 12·7-s − 8·17-s + 16·23-s − 4·25-s + 20·31-s − 8·41-s + 20·47-s + 68·49-s − 12·71-s − 16·73-s − 8·79-s − 16·97-s − 24·113-s − 96·119-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 192·161-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
| L(s) = 1 | + 4.53·7-s − 1.94·17-s + 3.33·23-s − 4/5·25-s + 3.59·31-s − 1.24·41-s + 2.91·47-s + 68/7·49-s − 1.42·71-s − 1.87·73-s − 0.900·79-s − 1.62·97-s − 2.25·113-s − 8.80·119-s + 1.81·121-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 + 15.1·161-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.25627426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.25627426\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 1866 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8618 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 7734 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 25866 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 276 T^{2} + 32522 T^{4} - 276 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.85202721364985541139427874737, −5.77032361049277956444920538168, −5.71306192164798241938418978514, −5.23058866513487587057606343349, −5.01733897881346374666627078596, −4.91620738448860821009956149025, −4.89490247468078222326062557204, −4.63245909141949251972767211876, −4.53652098156386998574919146442, −4.37510753131415477331869164057, −4.09520689296361647184295626239, −4.03467209628865535956530702021, −3.63274396101479278127290982258, −3.21904758878717879894230679270, −3.09020397533552441219803168236, −2.57863167042826641720332790134, −2.52948044814699542461169622185, −2.42705658725803742897242150660, −2.26659147025118220159764207602, −1.62958995975759591194702501662, −1.52999653407704602432605313168, −1.31508394645461363202472034148, −1.26030303638524620846714054568, −0.852313080381916879337510588779, −0.39414287635747183105339635758,
0.39414287635747183105339635758, 0.852313080381916879337510588779, 1.26030303638524620846714054568, 1.31508394645461363202472034148, 1.52999653407704602432605313168, 1.62958995975759591194702501662, 2.26659147025118220159764207602, 2.42705658725803742897242150660, 2.52948044814699542461169622185, 2.57863167042826641720332790134, 3.09020397533552441219803168236, 3.21904758878717879894230679270, 3.63274396101479278127290982258, 4.03467209628865535956530702021, 4.09520689296361647184295626239, 4.37510753131415477331869164057, 4.53652098156386998574919146442, 4.63245909141949251972767211876, 4.89490247468078222326062557204, 4.91620738448860821009956149025, 5.01733897881346374666627078596, 5.23058866513487587057606343349, 5.71306192164798241938418978514, 5.77032361049277956444920538168, 5.85202721364985541139427874737
