L(s) = 1 | + 3-s − 2·5-s + 9-s − 11-s − 6·13-s − 2·15-s + 2·17-s − 4·19-s − 4·23-s − 25-s + 27-s + 2·29-s − 4·31-s − 33-s − 2·37-s − 6·39-s + 10·41-s − 4·43-s − 2·45-s − 4·47-s − 7·49-s + 2·51-s + 6·53-s + 2·55-s − 4·57-s − 12·59-s − 6·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.174·33-s − 0.328·37-s − 0.960·39-s + 1.56·41-s − 0.609·43-s − 0.298·45-s − 0.583·47-s − 49-s + 0.280·51-s + 0.824·53-s + 0.269·55-s − 0.529·57-s − 1.56·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.530079082467179360986620811384, −8.548845995584116955684246785954, −7.73212630120393637746989050149, −7.35294342937331886949225371155, −6.15474901811631154449991906328, −4.91953811639053564159024850536, −4.15263903200078458209587475393, −3.10593494072283600488373899445, −2.04405336617156405213004194850, 0,
2.04405336617156405213004194850, 3.10593494072283600488373899445, 4.15263903200078458209587475393, 4.91953811639053564159024850536, 6.15474901811631154449991906328, 7.35294342937331886949225371155, 7.73212630120393637746989050149, 8.548845995584116955684246785954, 9.530079082467179360986620811384