L(s) = 1 | + 4-s − 2·9-s − 2·17-s + 2·25-s + 2·29-s − 2·36-s + 2·49-s − 4·53-s + 2·61-s − 64-s − 2·68-s + 81-s + 2·100-s − 2·101-s + 2·113-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 4-s − 2·9-s − 2·17-s + 2·25-s + 2·29-s − 2·36-s + 2·49-s − 4·53-s + 2·61-s − 64-s − 2·68-s + 81-s + 2·100-s − 2·101-s + 2·113-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6819051624\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6819051624\) |
\(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | | \( 1 \) |
good | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.033220507636001569377020174735, −7.50173042721762220985295565719, −7.11380780006616797039149253379, −7.01559081977551854781741787617, −6.76022627790139374268002653921, −6.67732483656634020697300820469, −6.52338077753115756403963101395, −6.12952934496471266055220023215, −5.88734466128779035642889046141, −5.71356026498068986380715965128, −5.60607804825502750833664267745, −4.95434884108137021917090736182, −4.83363210795382071037005506306, −4.78696513787114465596855014276, −4.42765494169131592724342385664, −4.16069121188017799988245779962, −3.69714944855823528783316673315, −3.32305686548256830028198989712, −3.08695566288795130996984523323, −2.80508990870431593667542644801, −2.54147346091484273542543942998, −2.45626233076909153801889847746, −1.95662763632202046635197634862, −1.53980948264443174890148283943, −0.852200475815374476339511689986,
0.852200475815374476339511689986, 1.53980948264443174890148283943, 1.95662763632202046635197634862, 2.45626233076909153801889847746, 2.54147346091484273542543942998, 2.80508990870431593667542644801, 3.08695566288795130996984523323, 3.32305686548256830028198989712, 3.69714944855823528783316673315, 4.16069121188017799988245779962, 4.42765494169131592724342385664, 4.78696513787114465596855014276, 4.83363210795382071037005506306, 4.95434884108137021917090736182, 5.60607804825502750833664267745, 5.71356026498068986380715965128, 5.88734466128779035642889046141, 6.12952934496471266055220023215, 6.52338077753115756403963101395, 6.67732483656634020697300820469, 6.76022627790139374268002653921, 7.01559081977551854781741787617, 7.11380780006616797039149253379, 7.50173042721762220985295565719, 8.033220507636001569377020174735