| L(s) = 1 | + 2-s − 3-s − 6-s − 8-s + 9-s − 2·11-s − 16-s − 2·17-s + 18-s − 2·22-s + 24-s − 25-s − 2·27-s + 2·33-s − 2·34-s − 41-s − 2·43-s + 48-s + 2·49-s − 50-s + 2·51-s − 2·54-s − 59-s + 64-s + 2·66-s − 67-s − 72-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 6-s − 8-s + 9-s − 2·11-s − 16-s − 2·17-s + 18-s − 2·22-s + 24-s − 25-s − 2·27-s + 2·33-s − 2·34-s − 41-s − 2·43-s + 48-s + 2·49-s − 50-s + 2·51-s − 2·54-s − 59-s + 64-s + 2·66-s − 67-s − 72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8340544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8340544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3193271450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3193271450\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.501136080335641113764785390223, −8.580028091241895276800817583341, −8.518084472217925575818419783092, −7.87831424170390515962659783632, −7.60600050595495350957093875903, −7.14063644780738912189150773393, −6.66308065809917527518560737244, −6.47111389318349038948290004592, −5.89848079136526335468119381254, −5.60294529812172190345556724898, −5.22681297632619147191299506100, −5.02092706577844635355845019888, −4.44488182282661863248818110047, −4.22222349083784377532298383670, −3.76693546526579314090881701803, −3.15503398177150434446826026739, −2.66637367385186739418632397740, −2.12332518763700143471709127563, −1.70433630472268427895816929752, −0.30056513078845589079011173225,
0.30056513078845589079011173225, 1.70433630472268427895816929752, 2.12332518763700143471709127563, 2.66637367385186739418632397740, 3.15503398177150434446826026739, 3.76693546526579314090881701803, 4.22222349083784377532298383670, 4.44488182282661863248818110047, 5.02092706577844635355845019888, 5.22681297632619147191299506100, 5.60294529812172190345556724898, 5.89848079136526335468119381254, 6.47111389318349038948290004592, 6.66308065809917527518560737244, 7.14063644780738912189150773393, 7.60600050595495350957093875903, 7.87831424170390515962659783632, 8.518084472217925575818419783092, 8.580028091241895276800817583341, 9.501136080335641113764785390223
