L(s) = 1 | + 12·7-s + 4·9-s + 12·23-s − 2·25-s − 24·31-s + 24·41-s − 12·47-s + 68·49-s + 48·63-s + 24·71-s − 16·73-s − 48·79-s + 6·81-s − 24·89-s + 16·97-s + 12·103-s − 24·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 144·161-s + 163-s + ⋯ |
L(s) = 1 | + 4.53·7-s + 4/3·9-s + 2.50·23-s − 2/5·25-s − 4.31·31-s + 3.74·41-s − 1.75·47-s + 68/7·49-s + 6.04·63-s + 2.84·71-s − 1.87·73-s − 5.40·79-s + 2/3·81-s − 2.54·89-s + 1.62·97-s + 1.18·103-s − 2.25·113-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 + 11.3·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.607815876\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.607815876\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2154 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 13002 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−8.262800374752680871730734032005, −8.260522407781695279026341291730, −7.81485946296382063015285684623, −7.71261604603941745553429114516, −7.58721003950842959170604791400, −7.28482561540555240027552186332, −7.03653098665293352722038615522, −6.91245819544501300939434490426, −6.57404807877663760465410882540, −5.76482792120193023122931491328, −5.56678722007914872470340498459, −5.55839792909619361598525833106, −5.40921528857901648583958074354, −4.74923742536354138952265870981, −4.71955098095264012975912458241, −4.46653452313539574964867108407, −4.30954755324240951155302494286, −3.97851902355099256593992072877, −3.44363050086677859955276502146, −3.10347785620104204351078802474, −2.31690934466756140279194415739, −2.20992760395063336859024110249, −1.54913821355366333262453587199, −1.44953091408692283366044663361, −1.22447303076733250374926377475,
1.22447303076733250374926377475, 1.44953091408692283366044663361, 1.54913821355366333262453587199, 2.20992760395063336859024110249, 2.31690934466756140279194415739, 3.10347785620104204351078802474, 3.44363050086677859955276502146, 3.97851902355099256593992072877, 4.30954755324240951155302494286, 4.46653452313539574964867108407, 4.71955098095264012975912458241, 4.74923742536354138952265870981, 5.40921528857901648583958074354, 5.55839792909619361598525833106, 5.56678722007914872470340498459, 5.76482792120193023122931491328, 6.57404807877663760465410882540, 6.91245819544501300939434490426, 7.03653098665293352722038615522, 7.28482561540555240027552186332, 7.58721003950842959170604791400, 7.71261604603941745553429114516, 7.81485946296382063015285684623, 8.260522407781695279026341291730, 8.262800374752680871730734032005