| L(s) = 1 | − 32·13-s + 10·25-s + 8·37-s + 196·49-s − 124·61-s − 296·73-s − 128·97-s + 388·109-s + 268·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.46·13-s + 2/5·25-s + 8/37·37-s + 4·49-s − 2.03·61-s − 4.05·73-s − 1.31·97-s + 3.55·109-s + 2.21·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.213·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.158021922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.158021922\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 533 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 587 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 1031 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 707 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 1742 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3158 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3986 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 5213 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5234 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 31 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 4118 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9974 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 1547 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 7703 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 11342 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{4} \) |
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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
−6.21895743555541327273323448301, −5.94145548463962808516832303388, −5.89927398324106580499741039614, −5.62517647176238080416208205804, −5.51245406969399305721300450409, −5.10895319406599392552444750615, −4.99257406280318301221567393053, −4.75198028039488891325858033454, −4.51717659651994571545481954558, −4.43699503185256766995682961944, −4.18981099596546276331650643148, −3.94377149640341345216257786874, −3.67479878854967632914320046839, −3.29334637581889975181995960688, −3.13028483026825790248267859379, −2.79940763442211789076747369219, −2.70864148126329691178979581925, −2.36514137020532690840035136381, −2.28264846135324403622180586183, −1.77987761084753340742937143830, −1.70625317469290917021147609729, −1.16017625810997476764188873012, −0.937447228610896350048525463048, −0.42948900672542783707939569905, −0.25796796134412198210283112372,
0.25796796134412198210283112372, 0.42948900672542783707939569905, 0.937447228610896350048525463048, 1.16017625810997476764188873012, 1.70625317469290917021147609729, 1.77987761084753340742937143830, 2.28264846135324403622180586183, 2.36514137020532690840035136381, 2.70864148126329691178979581925, 2.79940763442211789076747369219, 3.13028483026825790248267859379, 3.29334637581889975181995960688, 3.67479878854967632914320046839, 3.94377149640341345216257786874, 4.18981099596546276331650643148, 4.43699503185256766995682961944, 4.51717659651994571545481954558, 4.75198028039488891325858033454, 4.99257406280318301221567393053, 5.10895319406599392552444750615, 5.51245406969399305721300450409, 5.62517647176238080416208205804, 5.89927398324106580499741039614, 5.94145548463962808516832303388, 6.21895743555541327273323448301
