| L(s) = 1 | − 100·25-s − 110·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 130·169-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 + 241-s − 40·243-s + ⋯ |
| L(s) = 1 | − 20·25-s − 10·121-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 + 10·169-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 + 0.0644·241-s − 2.56·243-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 11^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20} \cdot 11^{20} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.003638257573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003638257573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + 20 T^{5} + p^{5} T^{10} )^{2} \) |
| 11 | \( ( 1 + p T^{2} )^{10} \) |
| 13 | \( ( 1 - p T^{2} )^{10} \) |
| good | 2 | \( ( 1 - 53 T^{10} + p^{10} T^{20} )^{2} \) |
| 5 | \( ( 1 + p T^{2} )^{20} \) |
| 7 | \( ( 1 - 1538 T^{10} + p^{10} T^{20} )^{2} \) |
| 17 | \( ( 1 + p T^{2} )^{20} \) |
| 19 | \( ( 1 - 4932794 T^{10} + p^{10} T^{20} )^{2} \) |
| 23 | \( ( 1 - 4728 T^{5} + p^{5} T^{10} )^{2}( 1 + 4728 T^{5} + p^{5} T^{10} )^{2} \) |
| 29 | \( ( 1 + p T^{2} )^{20} \) |
| 31 | \( ( 1 - p T^{2} )^{20} \) |
| 37 | \( ( 1 - p T^{2} )^{20} \) |
| 41 | \( ( 1 - 231619298 T^{10} + p^{10} T^{20} )^{2} \) |
| 43 | \( ( 1 - p T^{2} )^{20} \) |
| 47 | \( ( 1 + p T^{2} )^{20} \) |
| 53 | \( ( 1 - 24858 T^{5} + p^{5} T^{10} )^{2}( 1 + 24858 T^{5} + p^{5} T^{10} )^{2} \) |
| 59 | \( ( 1 + p T^{2} )^{20} \) |
| 61 | \( ( 1 - p T^{2} )^{20} \) |
| 67 | \( ( 1 - p T^{2} )^{20} \) |
| 71 | \( ( 1 + p T^{2} )^{20} \) |
| 73 | \( ( 1 - 3615121442 T^{10} + p^{10} T^{20} )^{2} \) |
| 79 | \( ( 1 - p T^{2} )^{20} \) |
| 83 | \( ( 1 - 7348587002 T^{10} + p^{10} T^{20} )^{2} \) |
| 89 | \( ( 1 + p T^{2} )^{20} \) |
| 97 | \( ( 1 - p T^{2} )^{20} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.51476651640923192001494385426, −2.43427347727883874059389538905, −2.42608125043533494531090970451, −2.32658732716410006696202494385, −2.26551577451527301017126456607, −2.23272531935447420161228503637, −2.20471163733515969035302258821, −2.03584748100889469080905177849, −2.03269741379948272648138893998, −2.02099039855821395437794812419, −1.90225603776380285744097007475, −1.86521175740659871520923468378, −1.75675770060327534331312720937, −1.71081775562577338368494593443, −1.52332293672219362873839315405, −1.47144112014320987995452947594, −1.38643573331377849453371839820, −1.37187230208963929327718755703, −1.18854439421693947988658326591, −1.11266214239977006598980489795, −0.66452940611962746271073072638, −0.55607445486716673832886439569, −0.27692121190235620558100467852, −0.07877063027308025270644558124, −0.05145938581735686734744354085,
0.05145938581735686734744354085, 0.07877063027308025270644558124, 0.27692121190235620558100467852, 0.55607445486716673832886439569, 0.66452940611962746271073072638, 1.11266214239977006598980489795, 1.18854439421693947988658326591, 1.37187230208963929327718755703, 1.38643573331377849453371839820, 1.47144112014320987995452947594, 1.52332293672219362873839315405, 1.71081775562577338368494593443, 1.75675770060327534331312720937, 1.86521175740659871520923468378, 1.90225603776380285744097007475, 2.02099039855821395437794812419, 2.03269741379948272648138893998, 2.03584748100889469080905177849, 2.20471163733515969035302258821, 2.23272531935447420161228503637, 2.26551577451527301017126456607, 2.32658732716410006696202494385, 2.42608125043533494531090970451, 2.43427347727883874059389538905, 2.51476651640923192001494385426
Plot not available for L-functions of degree greater than 10.
