| L(s) = 1 | − 3-s − 2·5-s + 3·7-s − 9-s − 3·11-s − 8·13-s + 2·15-s + 9·17-s − 3·19-s − 3·21-s − 4·23-s − 2·25-s + 5·27-s − 11·29-s + 5·31-s + 3·33-s − 6·35-s − 4·37-s + 8·39-s + 11·41-s − 4·43-s + 2·45-s + 2·47-s + 6·49-s − 9·51-s − 9·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.13·7-s − 1/3·9-s − 0.904·11-s − 2.21·13-s + 0.516·15-s + 2.18·17-s − 0.688·19-s − 0.654·21-s − 0.834·23-s − 2/5·25-s + 0.962·27-s − 2.04·29-s + 0.898·31-s + 0.522·33-s − 1.01·35-s − 0.657·37-s + 1.28·39-s + 1.71·41-s − 0.609·43-s + 0.298·45-s + 0.291·47-s + 6/7·49-s − 1.26·51-s − 1.23·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 2 T^{2} - 2 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 6 T^{2} + 6 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 26 T^{2} + 46 T^{3} + 26 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 50 T^{2} + 192 T^{3} + 50 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 9 T + 4 p T^{2} - 292 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 64 T^{2} + 180 T^{3} + 64 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 11 T + 102 T^{2} + 636 T^{3} + 102 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 5 T + 76 T^{2} - 314 T^{3} + 76 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 92 T^{2} + 246 T^{3} + 92 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 11 T + 110 T^{2} - 622 T^{3} + 110 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 37 T^{2} + 376 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 66 T^{2} - 372 T^{3} + 66 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 120 T^{2} + 632 T^{3} + 120 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 1308 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 26 T^{2} - 42 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 9 T + 218 T^{2} + 1192 T^{3} + 218 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 158 T^{2} + 58 T^{3} + 158 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 21 T + 302 T^{2} + 2764 T^{3} + 302 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 89 T^{2} - 68 T^{3} + 89 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 7 T + 204 T^{2} + 1062 T^{3} + 204 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 335 T^{2} + 3268 T^{3} + 335 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 28 T + 330 T^{2} - 2896 T^{3} + 330 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.34594761472028925664505815636, −7.32917433019601515780686071811, −6.98078304800888180086114323289, −6.32418828383105012292227726519, −6.06779315280294485683854622066, −6.05538046487009661587593475438, −6.00993816660285712310885482119, −5.45756285905518483488033473057, −5.30695977336905398355720072510, −5.14930638794445310424244747070, −4.97566284443671376326894139984, −4.54055392196831253313043191472, −4.52515640708979219378776444005, −4.12151404807576453470217569242, −4.10542585046753475281499223433, −3.63034305755035502871364709165, −3.15051257926425695128695552894, −3.10860984009089974773313640115, −2.96887781167001649968305514430, −2.36084623796459330576927140568, −2.28957320100449844668261461846, −1.96008575953867797219314147658, −1.55043815102810441727570080043, −1.23428361569132925743125456303, −0.882175080876133348166542961091, 0, 0, 0,
0.882175080876133348166542961091, 1.23428361569132925743125456303, 1.55043815102810441727570080043, 1.96008575953867797219314147658, 2.28957320100449844668261461846, 2.36084623796459330576927140568, 2.96887781167001649968305514430, 3.10860984009089974773313640115, 3.15051257926425695128695552894, 3.63034305755035502871364709165, 4.10542585046753475281499223433, 4.12151404807576453470217569242, 4.52515640708979219378776444005, 4.54055392196831253313043191472, 4.97566284443671376326894139984, 5.14930638794445310424244747070, 5.30695977336905398355720072510, 5.45756285905518483488033473057, 6.00993816660285712310885482119, 6.05538046487009661587593475438, 6.06779315280294485683854622066, 6.32418828383105012292227726519, 6.98078304800888180086114323289, 7.32917433019601515780686071811, 7.34594761472028925664505815636
