| L(s) = 1 | + 3·5-s + 5·7-s + 2·11-s + 2·17-s + 4·19-s + 7·23-s + 6·25-s − 7·29-s + 16·31-s + 15·35-s + 2·37-s − 41-s − 2·43-s + 13·47-s + 7·49-s − 10·53-s + 6·55-s + 6·59-s − 11·61-s − 67-s + 14·71-s + 16·73-s + 10·77-s + 6·79-s + 21·83-s + 6·85-s − 33·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.88·7-s + 0.603·11-s + 0.485·17-s + 0.917·19-s + 1.45·23-s + 6/5·25-s − 1.29·29-s + 2.87·31-s + 2.53·35-s + 0.328·37-s − 0.156·41-s − 0.304·43-s + 1.89·47-s + 49-s − 1.37·53-s + 0.809·55-s + 0.781·59-s − 1.40·61-s − 0.122·67-s + 1.66·71-s + 1.87·73-s + 1.13·77-s + 0.675·79-s + 2.30·83-s + 0.650·85-s − 3.49·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.81745378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.81745378\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 5 T + 18 T^{2} - 61 T^{3} + 18 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 25 T^{2} - 32 T^{3} + 25 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 15 T^{2} + 36 T^{3} + 15 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 15 T^{2} + 40 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 53 T^{2} - 148 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 18 T^{2} - 19 T^{3} + 18 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 82 T^{2} + 379 T^{3} + 82 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 16 T + 169 T^{2} - 1100 T^{3} + 169 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 103 T^{2} - 136 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + T + 114 T^{2} + 85 T^{3} + 114 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 2 T + 93 T^{2} + 64 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 13 T + 56 T^{2} - 99 T^{3} + 56 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 171 T^{2} + 1036 T^{3} + 171 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 117 T^{2} - 780 T^{3} + 117 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 11 T + 142 T^{2} + 823 T^{3} + 142 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + 68 T^{2} + 247 T^{3} + 68 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 14 T + 193 T^{2} - 1952 T^{3} + 193 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 3 p T^{2} - 1952 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 153 T^{2} - 1052 T^{3} + 153 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 21 T + 378 T^{2} - 3729 T^{3} + 378 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 - 30 T + 447 T^{2} - 4724 T^{3} + 447 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.26688442767580174298093648879, −6.66304164824867870511030635526, −6.62863916383555750764127922650, −6.37232033147249678088035253151, −6.17002753300069522167189671801, −5.87153827298815125076737466634, −5.66825751247638829937286973505, −5.16355035873671785504375497858, −5.14276142162837300309740621625, −5.12583897504809257764574604882, −4.69876454046187624165510210481, −4.55974308653066974228749603232, −4.25403695509309653207437152844, −3.87347737813239494746349380792, −3.54956488901264806253917341557, −3.38913500103024881152654747512, −2.87252023059663371208334437680, −2.71814308362277607916711297546, −2.52357339468073445710010036750, −1.91495136559814539427415706956, −1.84573268714590542162019713095, −1.61231915838136089027346665338, −1.09130790352822194774105716877, −0.821847915870118145921926751260, −0.72408876703583060903983536965,
0.72408876703583060903983536965, 0.821847915870118145921926751260, 1.09130790352822194774105716877, 1.61231915838136089027346665338, 1.84573268714590542162019713095, 1.91495136559814539427415706956, 2.52357339468073445710010036750, 2.71814308362277607916711297546, 2.87252023059663371208334437680, 3.38913500103024881152654747512, 3.54956488901264806253917341557, 3.87347737813239494746349380792, 4.25403695509309653207437152844, 4.55974308653066974228749603232, 4.69876454046187624165510210481, 5.12583897504809257764574604882, 5.14276142162837300309740621625, 5.16355035873671785504375497858, 5.66825751247638829937286973505, 5.87153827298815125076737466634, 6.17002753300069522167189671801, 6.37232033147249678088035253151, 6.62863916383555750764127922650, 6.66304164824867870511030635526, 7.26688442767580174298093648879
