L(s) = 1 | + 2·5-s + 2·11-s − 2·13-s + 4·17-s − 6·19-s + 2·25-s − 6·29-s + 16·31-s + 6·37-s − 10·43-s + 16·47-s + 10·49-s + 10·53-s + 4·55-s − 6·59-s − 18·61-s − 4·65-s + 10·67-s − 2·83-s + 8·85-s − 12·95-s − 4·97-s − 22·101-s − 14·107-s + 6·109-s + 12·113-s + 2·121-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.603·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s + 2/5·25-s − 1.11·29-s + 2.87·31-s + 0.986·37-s − 1.52·43-s + 2.33·47-s + 10/7·49-s + 1.37·53-s + 0.539·55-s − 0.781·59-s − 2.30·61-s − 0.496·65-s + 1.22·67-s − 0.219·83-s + 0.867·85-s − 1.23·95-s − 0.406·97-s − 2.18·101-s − 1.35·107-s + 0.574·109-s + 1.12·113-s + 2/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.236819119\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.236819119\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77311636803085939777919533946, −10.49457362522048707912170185274, −9.985492334308252459548449454653, −9.738002026339452232707527063661, −9.238224148555721332799289097895, −8.827648173254806009050575758447, −8.291743091132158793423402148583, −7.952217381380142089495914242612, −7.26456772043254885247206773098, −6.92280067119968710436887352894, −6.26462668120333481933310169658, −6.01316137671141209065033266010, −5.52372752609787966568192207742, −4.90345101020265287924670550326, −4.27159379593098255531242198801, −3.96234967114717235772572934758, −2.94986240719238815556388368069, −2.53296449977388423632414205611, −1.79537193214410255519390090922, −0.896756672785785301727100718539,
0.896756672785785301727100718539, 1.79537193214410255519390090922, 2.53296449977388423632414205611, 2.94986240719238815556388368069, 3.96234967114717235772572934758, 4.27159379593098255531242198801, 4.90345101020265287924670550326, 5.52372752609787966568192207742, 6.01316137671141209065033266010, 6.26462668120333481933310169658, 6.92280067119968710436887352894, 7.26456772043254885247206773098, 7.952217381380142089495914242612, 8.291743091132158793423402148583, 8.827648173254806009050575758447, 9.238224148555721332799289097895, 9.738002026339452232707527063661, 9.985492334308252459548449454653, 10.49457362522048707912170185274, 10.77311636803085939777919533946