L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 2·11-s + 15-s − 6·17-s − 6·19-s − 21-s + 8·23-s + 25-s + 27-s − 6·29-s + 6·31-s + 2·33-s − 35-s + 10·37-s + 2·41-s − 4·43-s + 45-s − 8·47-s + 49-s − 6·51-s + 12·53-s + 2·55-s − 6·57-s + 12·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.258·15-s − 1.45·17-s − 1.37·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.07·31-s + 0.348·33-s − 0.169·35-s + 1.64·37-s + 0.312·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.840·51-s + 1.64·53-s + 0.269·55-s − 0.794·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.609553046\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.609553046\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.142349484284735457798891984610, −7.10650208439281590456044121983, −6.64489974704019755292692350883, −6.08117998102279099664306761145, −5.01261837179861938646150173240, −4.32211417238723567906343380617, −3.60475664717797786552217750758, −2.58318227821274367317531752449, −2.04952008797739207737482098196, −0.801699931920923206500190982280,
0.801699931920923206500190982280, 2.04952008797739207737482098196, 2.58318227821274367317531752449, 3.60475664717797786552217750758, 4.32211417238723567906343380617, 5.01261837179861938646150173240, 6.08117998102279099664306761145, 6.64489974704019755292692350883, 7.10650208439281590456044121983, 8.142349484284735457798891984610