| L(s) = 1 | − 16·23-s − 16·25-s − 48·47-s + 12·49-s − 16·71-s + 32·73-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 3.33·23-s − 3.19·25-s − 7.00·47-s + 12/7·49-s − 1.89·71-s + 3.74·73-s + 1.09·121-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 + 3.38·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2131883542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2131883542\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−6.92635116847851327557431738941, −6.66945314300088502950600266534, −6.58655322798530919482034507423, −6.26569329355293482022834528538, −6.09036336712893615787529073628, −6.06875495580316535467887166582, −5.58880334786909656160455201104, −5.58277496465225715548969234861, −5.14231027992577088835262227129, −5.08527805937565944259596604056, −4.74022134228872666216830827641, −4.30907516834417058185510129630, −4.24127336391368014077821104667, −4.01888857842834514294148949867, −3.77173583554050019604924978975, −3.48279827679155516330239815175, −3.21268010215697667350033035799, −3.07420847649591753648099150545, −2.56028467783793664656114994770, −2.12096906952456972180466639581, −1.85357927114029764483781420041, −1.79382330855856724313872427603, −1.65203862045949977474044749941, −0.72099986513448904985670145152, −0.11657521055228975298798278440,
0.11657521055228975298798278440, 0.72099986513448904985670145152, 1.65203862045949977474044749941, 1.79382330855856724313872427603, 1.85357927114029764483781420041, 2.12096906952456972180466639581, 2.56028467783793664656114994770, 3.07420847649591753648099150545, 3.21268010215697667350033035799, 3.48279827679155516330239815175, 3.77173583554050019604924978975, 4.01888857842834514294148949867, 4.24127336391368014077821104667, 4.30907516834417058185510129630, 4.74022134228872666216830827641, 5.08527805937565944259596604056, 5.14231027992577088835262227129, 5.58277496465225715548969234861, 5.58880334786909656160455201104, 6.06875495580316535467887166582, 6.09036336712893615787529073628, 6.26569329355293482022834528538, 6.58655322798530919482034507423, 6.66945314300088502950600266534, 6.92635116847851327557431738941
