| L(s) = 1 | − 8·11-s + 48·31-s + 32·71-s + 28·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 | − 2.41·11-s + 8.62·31-s + 3.79·71-s + 28/9·81-s − 0.363·121-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 + 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1792373798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1792373798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 - 82 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 2 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 706 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 37 | \( ( 1 + 626 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 3682 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4382 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 + 7838 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 6722 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 12866 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−4.25421432713261118091130670994, −4.15439040531635122063195201315, −4.06446730031293273424176756314, −4.02043718534328130758109756937, −3.48358311244174988491866130082, −3.43711853741790695415959276300, −3.39851638833219968725678744720, −3.38264615662858754083214295274, −3.09313743157664001025871111908, −2.88761769751062528287664678723, −2.74326923170492608621234389814, −2.71648063391433995090061518703, −2.70401660553969385770566046344, −2.53528438795792660525817685387, −2.21995966185951167212032202380, −2.14861661200751618314124075784, −2.10207636133027307799557916694, −1.95371091646857377251063716970, −1.59910709128341406849560981367, −1.24998826971045197583405320262, −1.00177426588539700229442352985, −0.898042238476079977327665128164, −0.844201726264972490199061003636, −0.74869523348273858321809875535, −0.04404307938681357830875671089,
0.04404307938681357830875671089, 0.74869523348273858321809875535, 0.844201726264972490199061003636, 0.898042238476079977327665128164, 1.00177426588539700229442352985, 1.24998826971045197583405320262, 1.59910709128341406849560981367, 1.95371091646857377251063716970, 2.10207636133027307799557916694, 2.14861661200751618314124075784, 2.21995966185951167212032202380, 2.53528438795792660525817685387, 2.70401660553969385770566046344, 2.71648063391433995090061518703, 2.74326923170492608621234389814, 2.88761769751062528287664678723, 3.09313743157664001025871111908, 3.38264615662858754083214295274, 3.39851638833219968725678744720, 3.43711853741790695415959276300, 3.48358311244174988491866130082, 4.02043718534328130758109756937, 4.06446730031293273424176756314, 4.15439040531635122063195201315, 4.25421432713261118091130670994
Plot not available for L-functions of degree greater than 10.
