| L(s) = 1 | + 2.24·3-s − 1.47·5-s − 1.47·7-s + 2.05·9-s + 4.77·11-s − 4.49·13-s − 3.30·15-s + 1.71·17-s + 19-s − 3.30·21-s − 4·23-s − 2.83·25-s − 2.11·27-s − 7.80·29-s + 4.13·31-s + 10.7·33-s + 2.16·35-s − 3.30·37-s − 10.1·39-s − 9.80·41-s − 1.83·43-s − 3.02·45-s + 5.14·47-s − 4.83·49-s + 3.86·51-s + 12.8·53-s − 7.02·55-s + ⋯ |
| L(s) = 1 | + 1.29·3-s − 0.657·5-s − 0.555·7-s + 0.686·9-s + 1.44·11-s − 1.24·13-s − 0.854·15-s + 0.417·17-s + 0.229·19-s − 0.721·21-s − 0.834·23-s − 0.567·25-s − 0.407·27-s − 1.44·29-s + 0.742·31-s + 1.87·33-s + 0.365·35-s − 0.543·37-s − 1.62·39-s − 1.53·41-s − 0.280·43-s − 0.451·45-s + 0.750·47-s − 0.691·49-s + 0.541·51-s + 1.76·53-s − 0.947·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 + 1.47T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 - 1.71T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 4.13T + 31T^{2} \) |
| 37 | \( 1 + 3.30T + 37T^{2} \) |
| 41 | \( 1 + 9.80T + 41T^{2} \) |
| 43 | \( 1 + 1.83T + 43T^{2} \) |
| 47 | \( 1 - 5.14T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 9.05T + 59T^{2} \) |
| 61 | \( 1 + 8.91T + 61T^{2} \) |
| 67 | \( 1 + 5.92T + 67T^{2} \) |
| 71 | \( 1 - 6.05T + 71T^{2} \) |
| 73 | \( 1 + 0.335T + 73T^{2} \) |
| 79 | \( 1 - 0.366T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 - 2.36T + 89T^{2} \) |
| 97 | \( 1 + 7.05T + 97T^{2} \) |
| 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
−7.86077112719659031423192267719, −7.41523289040994427804144317935, −6.67557874845470258146178642500, −5.79901475292111110312918767390, −4.72358051663273923894233348380, −3.74120306229997364821711344697, −3.54124165814501002870000558131, −2.49637005079455046753922161169, −1.61902924040344362616824778446, 0,
1.61902924040344362616824778446, 2.49637005079455046753922161169, 3.54124165814501002870000558131, 3.74120306229997364821711344697, 4.72358051663273923894233348380, 5.79901475292111110312918767390, 6.67557874845470258146178642500, 7.41523289040994427804144317935, 7.86077112719659031423192267719
