| L(s) = 1 | + 2-s − 5-s − 8-s − 10-s + 13-s − 16-s + 3·17-s + 26-s + 29-s + 3·34-s + 37-s + 40-s + 3·41-s − 49-s + 58-s − 61-s + 64-s − 65-s − 2·73-s + 74-s + 80-s + 3·82-s − 3·85-s + 2·97-s − 98-s − 101-s − 104-s + ⋯ |
| L(s) = 1 | + 2-s − 5-s − 8-s − 10-s + 13-s − 16-s + 3·17-s + 26-s + 29-s + 3·34-s + 37-s + 40-s + 3·41-s − 49-s + 58-s − 61-s + 64-s − 65-s − 2·73-s + 74-s + 80-s + 3·82-s − 3·85-s + 2·97-s − 98-s − 101-s − 104-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.781198958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781198958\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−9.280718694452072487607390703817, −9.025782908602784649640022776771, −8.502398080140057413715759908144, −8.085041548680213529777037729218, −7.78692206718583662062925805682, −7.60547481548867515152192511153, −7.13017917365830523079803207826, −6.42968655845145281600919539763, −6.10067116961579229399085520087, −5.79587675306983398638962983987, −5.53384596024229564021296021648, −4.88935179742950353757681923528, −4.56049852078933499399715966805, −3.97793040974648648458047831785, −3.88033914378174634930021487110, −3.17487074632014835869300679392, −3.07251192133294447591483243248, −2.47612552298145770735485379354, −1.34424651680608372303898183835, −0.911119724019646971393717002791,
0.911119724019646971393717002791, 1.34424651680608372303898183835, 2.47612552298145770735485379354, 3.07251192133294447591483243248, 3.17487074632014835869300679392, 3.88033914378174634930021487110, 3.97793040974648648458047831785, 4.56049852078933499399715966805, 4.88935179742950353757681923528, 5.53384596024229564021296021648, 5.79587675306983398638962983987, 6.10067116961579229399085520087, 6.42968655845145281600919539763, 7.13017917365830523079803207826, 7.60547481548867515152192511153, 7.78692206718583662062925805682, 8.085041548680213529777037729218, 8.502398080140057413715759908144, 9.025782908602784649640022776771, 9.280718694452072487607390703817
