| L(s) = 1 | − 3-s + 4-s + 5-s + 11-s − 12-s − 15-s + 20-s + 27-s − 31-s − 33-s + 44-s − 3·47-s + 49-s + 55-s + 59-s − 60-s − 64-s − 3·67-s − 2·71-s − 81-s − 4·89-s + 93-s + 3·97-s + 3·103-s + 108-s + 3·113-s − 124-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 5-s + 11-s − 12-s − 15-s + 20-s + 27-s − 31-s − 33-s + 44-s − 3·47-s + 49-s + 55-s + 59-s − 60-s − 64-s − 3·67-s − 2·71-s − 81-s − 4·89-s + 93-s + 3·97-s + 3·103-s + 108-s + 3·113-s − 124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7979148883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7979148883\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.47304798123667485631526049744, −11.10637424347080435624991112781, −10.33397423621066693274945145815, −10.32021678039930942368899971606, −9.800076568314684370796295306076, −9.011073181409805392197870825483, −8.976788615723124782788477373738, −8.323244841692329338835330831455, −7.44708497976725443230297598546, −7.26147852179553981515788087811, −6.52516414814713817191214746369, −6.40366129546614905297319119607, −5.70933655233917327672443313745, −5.70625237338641607281202617609, −4.77124486338263749962979272486, −4.39025044858156413245126358183, −3.40871792634379354349638220839, −2.87883435466281258190426048627, −1.96041647102288231968680822271, −1.50337047905249870665130611832,
1.50337047905249870665130611832, 1.96041647102288231968680822271, 2.87883435466281258190426048627, 3.40871792634379354349638220839, 4.39025044858156413245126358183, 4.77124486338263749962979272486, 5.70625237338641607281202617609, 5.70933655233917327672443313745, 6.40366129546614905297319119607, 6.52516414814713817191214746369, 7.26147852179553981515788087811, 7.44708497976725443230297598546, 8.323244841692329338835330831455, 8.976788615723124782788477373738, 9.011073181409805392197870825483, 9.800076568314684370796295306076, 10.32021678039930942368899971606, 10.33397423621066693274945145815, 11.10637424347080435624991112781, 11.47304798123667485631526049744
