| L(s) = 1 | − 8·5-s + 12·11-s − 16-s − 12·23-s + 38·25-s + 8·31-s − 4·37-s + 4·47-s − 12·53-s − 96·55-s + 12·67-s − 32·71-s + 8·80-s − 28·97-s − 12·103-s − 4·113-s + 96·115-s + 86·121-s − 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·155-s + 157-s + ⋯ |
| L(s) = 1 | − 3.57·5-s + 3.61·11-s − 1/4·16-s − 2.50·23-s + 38/5·25-s + 1.43·31-s − 0.657·37-s + 0.583·47-s − 1.64·53-s − 12.9·55-s + 1.46·67-s − 3.79·71-s + 0.894·80-s − 2.84·97-s − 1.18·103-s − 0.376·113-s + 8.95·115-s + 7.81·121-s − 12.1·125-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 − 5.14·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.320280109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.320280109\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 1202 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 8542 T^{4} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 3374 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.17316585485065091683042287211, −6.97774526383859681997628035302, −6.71543793406511065003360313468, −6.59744012675904812153380726912, −6.47257877501886304466447080332, −6.12266888172147421944000213131, −5.77699668224538847010032137586, −5.69360503010811950539088849650, −5.36502973345812545763763445032, −4.63894298676932104818617427139, −4.61527827181284085608972464514, −4.44162268365540586227008271725, −4.41312005889468698818284349999, −4.05294688970705792632627900964, −3.73405418666427563633014942726, −3.68696290817587620437344966118, −3.63239748162628774196544867617, −3.06639863200297920068083556676, −2.88857339225610694836002488647, −2.63491077602132279163787352291, −1.72553210939248857473373519835, −1.70370220137157879971421784018, −1.32902108039810949861228467399, −0.64270358445023962313141979619, −0.43844413472104638530493798216,
0.43844413472104638530493798216, 0.64270358445023962313141979619, 1.32902108039810949861228467399, 1.70370220137157879971421784018, 1.72553210939248857473373519835, 2.63491077602132279163787352291, 2.88857339225610694836002488647, 3.06639863200297920068083556676, 3.63239748162628774196544867617, 3.68696290817587620437344966118, 3.73405418666427563633014942726, 4.05294688970705792632627900964, 4.41312005889468698818284349999, 4.44162268365540586227008271725, 4.61527827181284085608972464514, 4.63894298676932104818617427139, 5.36502973345812545763763445032, 5.69360503010811950539088849650, 5.77699668224538847010032137586, 6.12266888172147421944000213131, 6.47257877501886304466447080332, 6.59744012675904812153380726912, 6.71543793406511065003360313468, 6.97774526383859681997628035302, 7.17316585485065091683042287211
