| L(s) = 1 | + 6·2-s − 9·3-s + 24·4-s + 9·5-s − 54·6-s + 21·7-s + 80·8-s + 54·9-s + 54·10-s + 53·11-s − 216·12-s + 61·13-s + 126·14-s − 81·15-s + 240·16-s − 2·17-s + 324·18-s + 63·19-s + 216·20-s − 189·21-s + 318·22-s − 69·23-s − 720·24-s − 183·25-s + 366·26-s − 270·27-s + 504·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s + 0.804·5-s − 3.67·6-s + 1.13·7-s + 3.53·8-s + 2·9-s + 1.70·10-s + 1.45·11-s − 5.19·12-s + 1.30·13-s + 2.40·14-s − 1.39·15-s + 15/4·16-s − 0.0285·17-s + 4.24·18-s + 0.760·19-s + 2.41·20-s − 1.96·21-s + 3.08·22-s − 0.625·23-s − 6.12·24-s − 1.46·25-s + 2.76·26-s − 1.92·27-s + 3.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(31.23003832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(31.23003832\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 9 T + 264 T^{2} - 1618 T^{3} + 264 p^{3} T^{4} - 9 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 53 T + 252 p T^{2} - 70085 T^{3} + 252 p^{4} T^{4} - 53 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 61 T + 556 p T^{2} - 269856 T^{3} + 556 p^{4} T^{4} - 61 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 13528 T^{2} + 3628 T^{3} + 13528 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 63 T + 19804 T^{2} - 42965 p T^{3} + 19804 p^{3} T^{4} - 63 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 312 T + 74484 T^{2} - 15205550 T^{3} + 74484 p^{3} T^{4} - 312 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 220 T + 2270 p T^{2} - 12376550 T^{3} + 2270 p^{4} T^{4} - 220 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 44 T + 51720 T^{2} - 8908758 T^{3} + 51720 p^{3} T^{4} + 44 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 171 T + 2166 T^{2} + 7781985 T^{3} + 2166 p^{3} T^{4} - 171 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 627 T + 208138 T^{2} + 50157022 T^{3} + 208138 p^{3} T^{4} + 627 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 353 T + 31427 T^{2} + 11107654 T^{3} + 31427 p^{3} T^{4} - 353 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 504 T + 247801 T^{2} + 64056917 T^{3} + 247801 p^{3} T^{4} + 504 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 538 T + 404011 T^{2} - 177683293 T^{3} + 404011 p^{3} T^{4} - 538 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 256 T + 455793 T^{2} - 122355791 T^{3} + 455793 p^{3} T^{4} - 256 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 1113 T + 1084144 T^{2} - 613784866 T^{3} + 1084144 p^{3} T^{4} - 1113 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 53 T + 306434 T^{2} - 131428130 T^{3} + 306434 p^{3} T^{4} - 53 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 360 T + 703354 T^{2} - 82114836 T^{3} + 703354 p^{3} T^{4} - 360 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 134 T + 1029224 T^{2} - 177909718 T^{3} + 1029224 p^{3} T^{4} - 134 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1506 T + 2426596 T^{2} - 1829605402 T^{3} + 2426596 p^{3} T^{4} - 1506 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2381 T + 3379012 T^{2} - 3301721528 T^{3} + 3379012 p^{3} T^{4} - 2381 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 88 T + 2605532 T^{2} + 137721902 T^{3} + 2605532 p^{3} T^{4} + 88 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.446488715618765435219059020073, −7.958744088872059774805193424441, −7.77992650524409667239707981206, −7.77986730190477973009711503547, −6.93814474744904333447427221608, −6.79993553985949136435972669349, −6.67941682573029616787829204164, −6.16414980315537608916367697979, −6.12606743677945289657751299493, −6.06603659364882314503565473841, −5.40972937367743305560591128242, −5.36394958771885761063335484562, −5.04957028401576268301955165190, −4.52257131753370553412233964009, −4.49724108013594244841814373051, −4.29702838143686037608451591320, −3.55719272965556093408953065427, −3.44590392348046326276590869177, −3.38001188915950982271562424579, −2.26667085556188909894142384992, −2.01949348780647044692893375241, −1.99546644446973938367685688624, −1.14099143173492481957862385567, −1.03078390485532278505089932112, −0.69280061746187155936582715679,
0.69280061746187155936582715679, 1.03078390485532278505089932112, 1.14099143173492481957862385567, 1.99546644446973938367685688624, 2.01949348780647044692893375241, 2.26667085556188909894142384992, 3.38001188915950982271562424579, 3.44590392348046326276590869177, 3.55719272965556093408953065427, 4.29702838143686037608451591320, 4.49724108013594244841814373051, 4.52257131753370553412233964009, 5.04957028401576268301955165190, 5.36394958771885761063335484562, 5.40972937367743305560591128242, 6.06603659364882314503565473841, 6.12606743677945289657751299493, 6.16414980315537608916367697979, 6.67941682573029616787829204164, 6.79993553985949136435972669349, 6.93814474744904333447427221608, 7.77986730190477973009711503547, 7.77992650524409667239707981206, 7.958744088872059774805193424441, 8.446488715618765435219059020073
