| L(s) = 1 | − 3·2-s + 2·3-s − 4·5-s − 6·6-s − 2·7-s + 14·8-s + 12·10-s − 4·13-s + 6·14-s − 8·15-s − 21·16-s − 3·17-s − 8·19-s − 4·21-s + 2·23-s + 28·24-s + 8·25-s + 12·26-s + 2·27-s + 24·30-s − 18·31-s − 21·32-s + 9·34-s + 8·35-s + 4·37-s + 24·38-s − 8·39-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.15·3-s − 1.78·5-s − 2.44·6-s − 0.755·7-s + 4.94·8-s + 3.79·10-s − 1.10·13-s + 1.60·14-s − 2.06·15-s − 5.25·16-s − 0.727·17-s − 1.83·19-s − 0.872·21-s + 0.417·23-s + 5.71·24-s + 8/5·25-s + 2.35·26-s + 0.384·27-s + 4.38·30-s − 3.23·31-s − 3.71·32-s + 1.54·34-s + 1.35·35-s + 0.657·37-s + 3.89·38-s − 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{3} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_1$ | \( ( 1 + T )^{3} \) |
| 59 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 3 | $S_4\times C_2$ | \( 1 - 2 T + 4 T^{2} - 10 T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 14 T^{3} + 8 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 16 T^{2} + 26 T^{3} + 16 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 23 T^{2} + 48 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 72 T^{2} + 308 T^{3} + 72 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 49 T^{2} - 108 T^{3} + 49 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 44 T^{2} + 106 T^{3} + 44 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 18 T + 185 T^{2} + 1228 T^{3} + 185 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 31 T^{2} - 424 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 68 T^{2} - 242 T^{3} + 68 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 125 T^{2} + 500 T^{3} + 125 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 33 T^{2} - 324 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 28 T + 408 T^{2} + 3654 T^{3} + 408 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 123 T^{2} + 20 T^{3} + 123 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 105 T^{2} + 280 T^{3} + 105 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 22 T + 349 T^{2} + 3348 T^{3} + 349 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 199 T^{2} + 568 T^{3} + 199 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 56 T^{2} + 434 T^{3} + 56 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 185 T^{2} + 64 T^{3} + 185 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 20 T + 379 T^{2} + 3744 T^{3} + 379 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 375 T^{2} - 4300 T^{3} + 375 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.334182989338245451623834581613, −8.910861493498650870279590960295, −8.678047095075695089710895760373, −8.674507229020757055811726723855, −8.147329665285259247422953148926, −8.112841247579290778046672616941, −7.87698024028393778671239939590, −7.40702452866015130371296352644, −7.40034123956230788247987078711, −7.20223309657210519655909044921, −6.72334145877086288464852472797, −6.17601850343760958258734798693, −6.07889668494302932202845313808, −5.28643071391978720838586596986, −4.98896944906530395292962361871, −4.70476343882007130395904287321, −4.36838788236790300490124711263, −4.32568189369840918380567417881, −3.88028375575387295947059759194, −3.40307635245879645947400480037, −3.23473304251372872166886501377, −2.80658392677841449392391161211, −1.98381012843150294068347337598, −1.82389888864526036834252196026, −1.12796201658711298542593066247, 0, 0, 0,
1.12796201658711298542593066247, 1.82389888864526036834252196026, 1.98381012843150294068347337598, 2.80658392677841449392391161211, 3.23473304251372872166886501377, 3.40307635245879645947400480037, 3.88028375575387295947059759194, 4.32568189369840918380567417881, 4.36838788236790300490124711263, 4.70476343882007130395904287321, 4.98896944906530395292962361871, 5.28643071391978720838586596986, 6.07889668494302932202845313808, 6.17601850343760958258734798693, 6.72334145877086288464852472797, 7.20223309657210519655909044921, 7.40034123956230788247987078711, 7.40702452866015130371296352644, 7.87698024028393778671239939590, 8.112841247579290778046672616941, 8.147329665285259247422953148926, 8.674507229020757055811726723855, 8.678047095075695089710895760373, 8.910861493498650870279590960295, 9.334182989338245451623834581613
