| L(s) = 1 | − 3·3-s + 2·5-s − 4·7-s + 6·9-s − 3·11-s − 3·13-s − 6·15-s − 8·19-s + 12·21-s − 8·23-s − 7·25-s − 10·27-s + 14·29-s − 8·31-s + 9·33-s − 8·35-s − 8·37-s + 9·39-s + 14·41-s + 2·43-s + 12·45-s − 2·47-s − 49-s + 6·53-s − 6·55-s + 24·57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s − 1.51·7-s + 2·9-s − 0.904·11-s − 0.832·13-s − 1.54·15-s − 1.83·19-s + 2.61·21-s − 1.66·23-s − 7/5·25-s − 1.92·27-s + 2.59·29-s − 1.43·31-s + 1.56·33-s − 1.35·35-s − 1.31·37-s + 1.44·39-s + 2.18·41-s + 0.304·43-s + 1.78·45-s − 0.291·47-s − 1/7·49-s + 0.824·53-s − 0.809·55-s + 3.17·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 11^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 11^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9759428544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9759428544\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 2 T + 11 T^{2} - 18 T^{3} + 11 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 52 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 33 T^{2} + 26 T^{3} + 33 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 69 T^{2} + 292 T^{3} + 69 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 9 T^{2} - 100 T^{3} + 9 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 14 T + 73 T^{2} - 278 T^{3} + 73 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 109 T^{2} + 502 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 95 T^{2} + 560 T^{3} + 95 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 107 T^{2} - 600 T^{3} + 107 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 119 T^{2} - 170 T^{3} + 119 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 121 T^{2} + 164 T^{3} + 121 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 99 T^{2} - 292 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 896 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 39 T^{2} + 656 T^{3} + 39 p T^{4} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 117 T^{2} - 394 T^{3} + 117 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 12 p T^{3} + 65 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 99 T^{2} + 92 T^{3} + 99 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 87 T^{2} - 1334 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 209 T^{2} + 648 T^{3} + 209 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 151 T^{2} - 1670 T^{3} + 151 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 259 T^{2} + 680 T^{3} + 259 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.89379363077403344466526250559, −6.63933277786649454514259840440, −6.46487720176733677695027580595, −6.36544264603566894731199421431, −6.09689859792070929229096836577, −5.80239089512397967942352969981, −5.68133925693019263083372698064, −5.59860397212300859379392626756, −5.20642520461131002308077009481, −4.91321138720177896707357226828, −4.69119083206035792436040575864, −4.44121001512154020535944944240, −4.20608377779127112451299065294, −3.86768051503281397780665206693, −3.65092995283506945129181023883, −3.48886553691874850564137664180, −2.79485207100440291568929701163, −2.60954117344390265158089905053, −2.58986766705303208862154564028, −1.84719501908077670816957122330, −1.81576146418127408636679438527, −1.80433623712464973560688569973, −0.76868555394425876919103587583, −0.45035226438573655136915138272, −0.38943867373624245136402893805,
0.38943867373624245136402893805, 0.45035226438573655136915138272, 0.76868555394425876919103587583, 1.80433623712464973560688569973, 1.81576146418127408636679438527, 1.84719501908077670816957122330, 2.58986766705303208862154564028, 2.60954117344390265158089905053, 2.79485207100440291568929701163, 3.48886553691874850564137664180, 3.65092995283506945129181023883, 3.86768051503281397780665206693, 4.20608377779127112451299065294, 4.44121001512154020535944944240, 4.69119083206035792436040575864, 4.91321138720177896707357226828, 5.20642520461131002308077009481, 5.59860397212300859379392626756, 5.68133925693019263083372698064, 5.80239089512397967942352969981, 6.09689859792070929229096836577, 6.36544264603566894731199421431, 6.46487720176733677695027580595, 6.63933277786649454514259840440, 6.89379363077403344466526250559
