| L(s) = 1 | + 2-s + 5-s + 10-s + 11-s − 4·17-s + 22-s + 29-s − 31-s − 32-s − 4·34-s + 2·49-s + 55-s + 58-s + 4·61-s − 62-s − 64-s + 4·67-s + 71-s + 83-s − 4·85-s + 2·98-s + 101-s − 109-s + 110-s + 113-s + 4·122-s + 127-s + ⋯ |
| L(s) = 1 | + 2-s + 5-s + 10-s + 11-s − 4·17-s + 22-s + 29-s − 31-s − 32-s − 4·34-s + 2·49-s + 55-s + 58-s + 4·61-s − 62-s − 64-s + 4·67-s + 71-s + 83-s − 4·85-s + 2·98-s + 101-s − 109-s + 110-s + 113-s + 4·122-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4626801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4626801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.176015909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.176015909\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 239 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 31 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$ | \( ( 1 - T )^{4} \) |
| 67 | $C_1$ | \( ( 1 - T )^{4} \) |
| 71 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 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
−9.277357265970803211448541053451, −9.171472081492781686708045463320, −8.659267805538340213887311995116, −8.550239327567276490040822434967, −8.023186007479946412827564376962, −7.21688393839495074831174320090, −6.86465073720404434702563489460, −6.63581507066028062088981321946, −6.49758097879193600084400970858, −5.83005356783556617920389413494, −5.35380166455750484776377771751, −5.10119724032208083074151577129, −4.58115308034155493248053161048, −4.21566624381821295087942567330, −3.83251471585125388032081975924, −3.60799011593402461985117493752, −2.38727750855184818971121654752, −2.24237372544536865240578055503, −2.05051819705335786598666126915, −0.893201013245216599471994312764,
0.893201013245216599471994312764, 2.05051819705335786598666126915, 2.24237372544536865240578055503, 2.38727750855184818971121654752, 3.60799011593402461985117493752, 3.83251471585125388032081975924, 4.21566624381821295087942567330, 4.58115308034155493248053161048, 5.10119724032208083074151577129, 5.35380166455750484776377771751, 5.83005356783556617920389413494, 6.49758097879193600084400970858, 6.63581507066028062088981321946, 6.86465073720404434702563489460, 7.21688393839495074831174320090, 8.023186007479946412827564376962, 8.550239327567276490040822434967, 8.659267805538340213887311995116, 9.171472081492781686708045463320, 9.277357265970803211448541053451
