| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 9-s − 11-s + 2·13-s − 16-s + 17-s − 18-s + 22-s + 2·23-s + 24-s − 25-s − 2·26-s + 2·27-s − 2·31-s − 33-s − 34-s + 2·39-s − 2·46-s − 48-s + 50-s + 51-s + 53-s − 2·54-s + 2·62-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 9-s − 11-s + 2·13-s − 16-s + 17-s − 18-s + 22-s + 2·23-s + 24-s − 25-s − 2·26-s + 2·27-s − 2·31-s − 33-s − 34-s + 2·39-s − 2·46-s − 48-s + 50-s + 51-s + 53-s − 2·54-s + 2·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.358433711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358433711\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.773532678547593282894925144079, −8.746300465181569890677185170597, −8.344381114951706455960692511036, −7.993655932730052409396649925737, −7.67619608016697543773836867281, −7.22151741026484969351553466016, −7.08952523983129606988435928847, −6.53859165999034129900148501337, −6.05146604952542531929941469898, −5.49951932307031143968788975978, −5.22754437550127131603693896878, −4.82767970733724140174945465366, −4.22503186318174153264279351925, −3.70191053133206924462868227875, −3.57395742603145474214721015976, −3.00899310910461612365730960062, −2.48310011523602275638513868518, −1.87499124814292530563992315569, −1.30549339138968095083999445735, −0.915565441103698760847667141649,
0.915565441103698760847667141649, 1.30549339138968095083999445735, 1.87499124814292530563992315569, 2.48310011523602275638513868518, 3.00899310910461612365730960062, 3.57395742603145474214721015976, 3.70191053133206924462868227875, 4.22503186318174153264279351925, 4.82767970733724140174945465366, 5.22754437550127131603693896878, 5.49951932307031143968788975978, 6.05146604952542531929941469898, 6.53859165999034129900148501337, 7.08952523983129606988435928847, 7.22151741026484969351553466016, 7.67619608016697543773836867281, 7.993655932730052409396649925737, 8.344381114951706455960692511036, 8.746300465181569890677185170597, 8.773532678547593282894925144079
