L(s) = 1 | − 48·19-s − 4·25-s − 32·49-s + 80·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 80·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
L(s) = 1 | − 11.0·19-s − 4/5·25-s − 4.57·49-s + 7.27·121-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 − 6.15·169-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.578019862\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.578019862\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
good | 7 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 6 T + p T^{2} )^{8} \) |
| 23 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 67 | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 14 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.75829877363639456715144667288, −3.51571391005420820129309732612, −3.43689164368790767464992377549, −3.37605512447062917723233081208, −3.29114422296748561612133504504, −3.12963567901789179942801515866, −2.68855097424618587932169025292, −2.64564529203043243529829044961, −2.56385910816753463458749609008, −2.54386871396920340085228706424, −2.37392418908361656493572057160, −2.27668675301760945203107859196, −2.23142797406023346227331038619, −2.10128598704338118334297723476, −1.81553402634386337737615904284, −1.75043768036228261627317813180, −1.72121674956024784836165101108, −1.51415824322278339206177223961, −1.50393779136450681226316940444, −1.37371967150443383049790152902, −0.855864800327848762963349556569, −0.39298305501320008442952426042, −0.37597171339505609728724607759, −0.35256959671969782372835165590, −0.28496616764813339341146801709,
0.28496616764813339341146801709, 0.35256959671969782372835165590, 0.37597171339505609728724607759, 0.39298305501320008442952426042, 0.855864800327848762963349556569, 1.37371967150443383049790152902, 1.50393779136450681226316940444, 1.51415824322278339206177223961, 1.72121674956024784836165101108, 1.75043768036228261627317813180, 1.81553402634386337737615904284, 2.10128598704338118334297723476, 2.23142797406023346227331038619, 2.27668675301760945203107859196, 2.37392418908361656493572057160, 2.54386871396920340085228706424, 2.56385910816753463458749609008, 2.64564529203043243529829044961, 2.68855097424618587932169025292, 3.12963567901789179942801515866, 3.29114422296748561612133504504, 3.37605512447062917723233081208, 3.43689164368790767464992377549, 3.51571391005420820129309732612, 3.75829877363639456715144667288
Plot not available for L-functions of degree greater than 10.