L(s) = 1 | − 3-s − 5-s − 7-s + 11-s − 2·13-s + 15-s + 4·17-s + 21-s + 27-s + 29-s − 33-s + 35-s + 2·39-s + 47-s − 4·51-s − 55-s + 2·65-s − 2·71-s − 2·73-s − 77-s + 79-s − 81-s + 83-s − 4·85-s − 87-s + 2·91-s + 97-s + ⋯ |
L(s) = 1 | − 3-s − 5-s − 7-s + 11-s − 2·13-s + 15-s + 4·17-s + 21-s + 27-s + 29-s − 33-s + 35-s + 2·39-s + 47-s − 4·51-s − 55-s + 2·65-s − 2·71-s − 2·73-s − 77-s + 79-s − 81-s + 83-s − 4·85-s − 87-s + 2·91-s + 97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4967989111\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4967989111\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
good | 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−10.06730644495767293612875970793, −9.893887134316449520003900315330, −9.348551761298826515776756762890, −9.041205284071703107447268688878, −8.315673376621380181281186083999, −7.959218325176305146960312070330, −7.54736716130597860415563334494, −7.20111279648771999587577018523, −6.98078418398655709014225964889, −6.12070785493283817583891144665, −6.00862089402620994796524798995, −5.57621709765312417170182597516, −4.86853994502728291245196181917, −4.83677616947501216250468290792, −3.91042395338467591995970446413, −3.60742640802207272197799622227, −2.97895615242959328264702728955, −2.75524952987346086353900111449, −1.45142171725875049176399491500, −0.70555636344158107069093094353,
0.70555636344158107069093094353, 1.45142171725875049176399491500, 2.75524952987346086353900111449, 2.97895615242959328264702728955, 3.60742640802207272197799622227, 3.91042395338467591995970446413, 4.83677616947501216250468290792, 4.86853994502728291245196181917, 5.57621709765312417170182597516, 6.00862089402620994796524798995, 6.12070785493283817583891144665, 6.98078418398655709014225964889, 7.20111279648771999587577018523, 7.54736716130597860415563334494, 7.959218325176305146960312070330, 8.315673376621380181281186083999, 9.041205284071703107447268688878, 9.348551761298826515776756762890, 9.893887134316449520003900315330, 10.06730644495767293612875970793