L(s) = 1 | − 5-s + 4·11-s − 2·13-s + 2·17-s − 8·19-s − 4·23-s + 25-s − 6·29-s − 2·37-s + 6·41-s − 4·43-s + 12·47-s − 7·49-s − 6·53-s − 4·55-s + 12·59-s − 14·61-s + 2·65-s + 12·67-s + 2·73-s − 8·79-s − 4·83-s − 2·85-s − 2·89-s + 8·95-s − 14·97-s − 14·101-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 1.83·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.328·37-s + 0.937·41-s − 0.609·43-s + 1.75·47-s − 49-s − 0.824·53-s − 0.539·55-s + 1.56·59-s − 1.79·61-s + 0.248·65-s + 1.46·67-s + 0.234·73-s − 0.900·79-s − 0.439·83-s − 0.216·85-s − 0.211·89-s + 0.820·95-s − 1.42·97-s − 1.39·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.376551830272548512592493257349, −7.69167978948822675963723426669, −6.83877914599489268844584933414, −6.22381565074983412013957302616, −5.31151708115231160045329698021, −4.18337696150292350218664744241, −3.86368157122591542985344652671, −2.56873695382649773958744744086, −1.52992113126652069198507210517, 0,
1.52992113126652069198507210517, 2.56873695382649773958744744086, 3.86368157122591542985344652671, 4.18337696150292350218664744241, 5.31151708115231160045329698021, 6.22381565074983412013957302616, 6.83877914599489268844584933414, 7.69167978948822675963723426669, 8.376551830272548512592493257349