| L(s) = 1 | − 6·3-s + 21·9-s + 18·11-s − 12·19-s + 2·25-s − 54·27-s − 108·33-s + 18·41-s − 18·43-s − 14·49-s + 72·57-s + 18·59-s + 6·67-s + 20·73-s − 12·75-s + 108·81-s − 36·83-s + 2·97-s + 378·99-s + 48·113-s + 167·121-s − 108·123-s + 127-s + 108·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 7·9-s + 5.42·11-s − 2.75·19-s + 2/5·25-s − 10.3·27-s − 18.8·33-s + 2.81·41-s − 2.74·43-s − 2·49-s + 9.53·57-s + 2.34·59-s + 0.733·67-s + 2.34·73-s − 1.38·75-s + 12·81-s − 3.95·83-s + 0.203·97-s + 37.9·99-s + 4.51·113-s + 15.1·121-s − 9.73·123-s + 0.0887·127-s + 9.50·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.389846264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389846264\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 2 T^{2} - 165 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 26 T^{2} + 147 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 26 T^{2} - 165 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 22 T^{2} - 1725 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 9 T + 86 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^3$ | \( 1 + 26 T^{2} - 3045 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 83 | $C_2^2$ | \( ( 1 + 18 T + 191 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.86225578026672716363170686607, −6.58260913487890575893657817418, −6.57666144807762741697190410026, −6.26692934233575124946294673449, −6.25184919060568999398418839300, −6.01594673829508646436149777431, −5.86585160786309311148241300211, −5.41702532630897630493696247525, −5.40767423827741668140545946447, −4.92789180250733674211606388388, −4.59994272175552682678297190532, −4.46578859350725885778049819793, −4.45056448115224246455841261352, −4.10850875392062440669071888730, −3.87586365739569450190754808460, −3.73620157241237739890366136679, −3.55806975267846701759229156315, −3.02953005790363397746601176677, −2.48105791744591177752966204221, −1.77797244456041123063065437449, −1.72440196015286130785042541475, −1.67993613006641959013621805180, −1.06250292230322306631898930342, −0.801670029554136078794770375776, −0.47415681326345612261619121912,
0.47415681326345612261619121912, 0.801670029554136078794770375776, 1.06250292230322306631898930342, 1.67993613006641959013621805180, 1.72440196015286130785042541475, 1.77797244456041123063065437449, 2.48105791744591177752966204221, 3.02953005790363397746601176677, 3.55806975267846701759229156315, 3.73620157241237739890366136679, 3.87586365739569450190754808460, 4.10850875392062440669071888730, 4.45056448115224246455841261352, 4.46578859350725885778049819793, 4.59994272175552682678297190532, 4.92789180250733674211606388388, 5.40767423827741668140545946447, 5.41702532630897630493696247525, 5.86585160786309311148241300211, 6.01594673829508646436149777431, 6.25184919060568999398418839300, 6.26692934233575124946294673449, 6.57666144807762741697190410026, 6.58260913487890575893657817418, 6.86225578026672716363170686607
