| L(s) = 1 | + 3-s − 5-s − 0.561·7-s + 9-s − 2·11-s − 2·13-s − 15-s − 2.56·17-s + 1.12·19-s − 0.561·21-s + 23-s + 25-s + 27-s + 4.56·29-s + 3.68·31-s − 2·33-s + 0.561·35-s + 3.68·37-s − 2·39-s + 5.68·41-s − 45-s + 10.2·47-s − 6.68·49-s − 2.56·51-s − 3.43·53-s + 2·55-s + 1.12·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.212·7-s + 0.333·9-s − 0.603·11-s − 0.554·13-s − 0.258·15-s − 0.621·17-s + 0.257·19-s − 0.122·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.847·29-s + 0.661·31-s − 0.348·33-s + 0.0949·35-s + 0.605·37-s − 0.320·39-s + 0.887·41-s − 0.149·45-s + 1.49·47-s − 0.954·49-s − 0.358·51-s − 0.472·53-s + 0.269·55-s + 0.148·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 - 3.68T + 37T^{2} \) |
| 41 | \( 1 - 5.68T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 3.43T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 5.68T + 83T^{2} \) |
| 89 | \( 1 + 6.24T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.73323728027150846939065942207, −7.30707288034299648235951423277, −6.45430427703659402117185138550, −5.65903621050684819637864559100, −4.60432671217303651691664758275, −4.25152857906856567674211074136, −2.96711207265964263190159840981, −2.69327730517979416106497424198, −1.37801691305580559554672297735, 0,
1.37801691305580559554672297735, 2.69327730517979416106497424198, 2.96711207265964263190159840981, 4.25152857906856567674211074136, 4.60432671217303651691664758275, 5.65903621050684819637864559100, 6.45430427703659402117185138550, 7.30707288034299648235951423277, 7.73323728027150846939065942207
