| L(s) = 1 | − 20·41-s − 12·49-s + 40·61-s − 20·89-s + 8·101-s − 24·109-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 3.12·41-s − 1.71·49-s + 5.12·61-s − 2.11·89-s + 0.796·101-s − 2.29·109-s − 3.09·121-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 + 1.53·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3305823825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3305823825\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 129 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.42203298465228833083737670731, −5.39784149886645335350533202719, −5.26463914749291873568210757351, −5.01829961303281057425261312630, −4.83719862463972772640668932897, −4.82164592941250853059079769600, −4.29865035123383011571593854429, −4.26128975382218267191807533246, −3.99259519659740165898230981111, −3.91440677537429529658819380054, −3.65561792616361294338861589601, −3.44713520898222059297840299683, −3.26124728183219900421952942671, −3.12631947315584158575051198782, −2.85720539381368682477026792647, −2.46638329762792549391810432918, −2.35704000147278000255294995888, −2.30861678740120204224271111606, −1.88520164164897844474793435139, −1.73422867820027683031652218276, −1.29348053404205404734656024124, −1.12233395391731144027394747420, −1.12130258632548584779506121241, −0.43061105971833427041287858923, −0.087654335613115036668467431522,
0.087654335613115036668467431522, 0.43061105971833427041287858923, 1.12130258632548584779506121241, 1.12233395391731144027394747420, 1.29348053404205404734656024124, 1.73422867820027683031652218276, 1.88520164164897844474793435139, 2.30861678740120204224271111606, 2.35704000147278000255294995888, 2.46638329762792549391810432918, 2.85720539381368682477026792647, 3.12631947315584158575051198782, 3.26124728183219900421952942671, 3.44713520898222059297840299683, 3.65561792616361294338861589601, 3.91440677537429529658819380054, 3.99259519659740165898230981111, 4.26128975382218267191807533246, 4.29865035123383011571593854429, 4.82164592941250853059079769600, 4.83719862463972772640668932897, 5.01829961303281057425261312630, 5.26463914749291873568210757351, 5.39784149886645335350533202719, 5.42203298465228833083737670731
