| L(s) = 1 | + 4-s − 2·7-s − 2·28-s + 2·37-s + 2·43-s + 3·49-s − 64-s − 2·67-s − 2·79-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4-s − 2·7-s − 2·28-s + 2·37-s + 2·43-s + 3·49-s − 64-s − 2·67-s − 2·79-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.059640727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059640727\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−9.617057177745172190576011958428, −9.584393752283810721806893958335, −9.008455228871833970233088811475, −8.884379219407939307763741609540, −8.152954054950129406405947759117, −7.62155594495233913214084869127, −7.42001928227150077772234916988, −6.83692222167584058846613281271, −6.75890356213655413725298278223, −6.08152342081798955968313315436, −5.86429023127124861125893333481, −5.72027586380494215003391839695, −4.74124173334812975319517000054, −4.24112695776208105432444427395, −3.95235208207629412105744481474, −3.03726286108766613051093752769, −3.00540177480996256264934509273, −2.48516409891222525293330258201, −1.82560264951628206134338892523, −0.804016494520782995637871771400,
0.804016494520782995637871771400, 1.82560264951628206134338892523, 2.48516409891222525293330258201, 3.00540177480996256264934509273, 3.03726286108766613051093752769, 3.95235208207629412105744481474, 4.24112695776208105432444427395, 4.74124173334812975319517000054, 5.72027586380494215003391839695, 5.86429023127124861125893333481, 6.08152342081798955968313315436, 6.75890356213655413725298278223, 6.83692222167584058846613281271, 7.42001928227150077772234916988, 7.62155594495233913214084869127, 8.152954054950129406405947759117, 8.884379219407939307763741609540, 9.008455228871833970233088811475, 9.584393752283810721806893958335, 9.617057177745172190576011958428
