| L(s) = 1 | − 3·8-s + 30·25-s − 4·27-s + 42·49-s − 72·59-s + 64-s + 36·101-s + 66·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 90·200-s + 211-s + 12·216-s + ⋯ |
| L(s) = 1 | − 1.06·8-s + 6·25-s − 0.769·27-s + 6·49-s − 9.37·59-s + 1/8·64-s + 3.58·101-s + 6·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s − 6.36·200-s + 0.0688·211-s + 0.816·216-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.448720197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.448720197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3 T^{3} + p^{3} T^{6} \) |
| 3 | \( 1 + 4 T^{3} + p^{3} T^{6} \) |
| 23 | \( ( 1 + p T^{2} )^{3} \) |
| good | 5 | \( ( 1 - p T^{2} )^{6} \) |
| 7 | \( ( 1 - p T^{2} )^{6} \) |
| 11 | \( ( 1 - p T^{2} )^{6} \) |
| 13 | \( ( 1 - 74 T^{3} + p^{3} T^{6} )( 1 + 74 T^{3} + p^{3} T^{6} ) \) |
| 17 | \( ( 1 + p T^{2} )^{6} \) |
| 19 | \( ( 1 + p T^{2} )^{6} \) |
| 29 | \( ( 1 - 282 T^{3} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 344 T^{3} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + p T^{2} )^{6} \) |
| 41 | \( ( 1 - 426 T^{3} + p^{3} T^{6} )( 1 + 426 T^{3} + p^{3} T^{6} ) \) |
| 43 | \( ( 1 + p T^{2} )^{6} \) |
| 47 | \( ( 1 - 48 T^{3} + p^{3} T^{6} )( 1 + 48 T^{3} + p^{3} T^{6} ) \) |
| 53 | \( ( 1 - p T^{2} )^{6} \) |
| 59 | \( ( 1 + 12 T + p T^{2} )^{6} \) |
| 61 | \( ( 1 + p T^{2} )^{6} \) |
| 67 | \( ( 1 + p T^{2} )^{6} \) |
| 71 | \( ( 1 - 1176 T^{3} + p^{3} T^{6} )( 1 + 1176 T^{3} + p^{3} T^{6} ) \) |
| 73 | \( ( 1 + 1226 T^{3} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 - p T^{2} )^{6} \) |
| 83 | \( ( 1 - p T^{2} )^{6} \) |
| 89 | \( ( 1 + p T^{2} )^{6} \) |
| 97 | \( ( 1 - p T^{2} )^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.89919609384879735731607621384, −5.55768117179082285025276396946, −5.42172844991354223041747103654, −5.33345821022543083031021657296, −5.29777537608637311753708079234, −4.62079357500161832176779787366, −4.61106166882319249413920561198, −4.59225767766055880841198892869, −4.47611547014437323758337970922, −4.45076735910433631161509513338, −4.21735613640730584453813483501, −3.54130939077545999881427635171, −3.44491113404358681511823150114, −3.27659904965861827877107494492, −3.20810089349670382316396145507, −3.12123940796044976398457012847, −2.82479845461763649290073598976, −2.54522710359210399171294487353, −2.45190927733745579228604031249, −2.01080321578425073697671805389, −1.88014966623508339697270848334, −1.30898193353791838536453978326, −1.21276530993520022855550694278, −0.74690058599252993485919153500, −0.51205747796455015544015488707,
0.51205747796455015544015488707, 0.74690058599252993485919153500, 1.21276530993520022855550694278, 1.30898193353791838536453978326, 1.88014966623508339697270848334, 2.01080321578425073697671805389, 2.45190927733745579228604031249, 2.54522710359210399171294487353, 2.82479845461763649290073598976, 3.12123940796044976398457012847, 3.20810089349670382316396145507, 3.27659904965861827877107494492, 3.44491113404358681511823150114, 3.54130939077545999881427635171, 4.21735613640730584453813483501, 4.45076735910433631161509513338, 4.47611547014437323758337970922, 4.59225767766055880841198892869, 4.61106166882319249413920561198, 4.62079357500161832176779787366, 5.29777537608637311753708079234, 5.33345821022543083031021657296, 5.42172844991354223041747103654, 5.55768117179082285025276396946, 5.89919609384879735731607621384
Plot not available for L-functions of degree greater than 10.
