L(s) = 1 | + 2·5-s + 4·11-s + 13-s − 17-s + 4·19-s − 25-s + 2·29-s + 8·31-s − 2·37-s − 2·41-s + 4·43-s + 8·47-s − 7·49-s + 10·53-s + 8·55-s + 4·59-s + 14·61-s + 2·65-s + 4·67-s − 14·73-s + 8·79-s − 4·83-s − 2·85-s + 6·89-s + 8·95-s − 6·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.20·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s − 49-s + 1.37·53-s + 1.07·55-s + 0.520·59-s + 1.79·61-s + 0.248·65-s + 0.488·67-s − 1.63·73-s + 0.900·79-s − 0.439·83-s − 0.216·85-s + 0.635·89-s + 0.820·95-s − 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.700542831\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.700542831\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.00775514696786, −14.31796887385678, −14.11215667428918, −13.44096808839433, −13.20703747705707, −12.32573073063147, −11.77988909310708, −11.56784385709540, −10.67489202731927, −10.20508497309687, −9.602644429989829, −9.285749222231189, −8.580362947631242, −8.137424458555636, −7.219381211372168, −6.770376828847969, −6.204992163634711, −5.651250246475473, −5.084028941580542, −4.234052196545388, −3.760136957911145, −2.881002245172081, −2.228547499450869, −1.400180694780998, −0.7962902065217460,
0.7962902065217460, 1.400180694780998, 2.228547499450869, 2.881002245172081, 3.760136957911145, 4.234052196545388, 5.084028941580542, 5.651250246475473, 6.204992163634711, 6.770376828847969, 7.219381211372168, 8.137424458555636, 8.580362947631242, 9.285749222231189, 9.602644429989829, 10.20508497309687, 10.67489202731927, 11.56784385709540, 11.77988909310708, 12.32573073063147, 13.20703747705707, 13.44096808839433, 14.11215667428918, 14.31796887385678, 15.00775514696786