L(s) = 1 | − 2-s + 3-s + 4-s + 1.21·5-s − 6-s − 1.75·7-s − 8-s + 9-s − 1.21·10-s + 11-s + 12-s + 3.89·13-s + 1.75·14-s + 1.21·15-s + 16-s − 7.19·17-s − 18-s − 3.80·19-s + 1.21·20-s − 1.75·21-s − 22-s − 4.79·23-s − 24-s − 3.51·25-s − 3.89·26-s + 27-s − 1.75·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.545·5-s − 0.408·6-s − 0.661·7-s − 0.353·8-s + 0.333·9-s − 0.385·10-s + 0.301·11-s + 0.288·12-s + 1.07·13-s + 0.467·14-s + 0.314·15-s + 0.250·16-s − 1.74·17-s − 0.235·18-s − 0.872·19-s + 0.272·20-s − 0.382·21-s − 0.213·22-s − 0.999·23-s − 0.204·24-s − 0.702·25-s − 0.763·26-s + 0.192·27-s − 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.712621932\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.712621932\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 - 1.21T + 5T^{2} \) |
| 7 | \( 1 + 1.75T + 7T^{2} \) |
| 13 | \( 1 - 3.89T + 13T^{2} \) |
| 17 | \( 1 + 7.19T + 17T^{2} \) |
| 19 | \( 1 + 3.80T + 19T^{2} \) |
| 23 | \( 1 + 4.79T + 23T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 - 8.01T + 31T^{2} \) |
| 37 | \( 1 - 7.80T + 37T^{2} \) |
| 41 | \( 1 - 0.834T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 - 7.11T + 53T^{2} \) |
| 59 | \( 1 - 4.02T + 59T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 4.50T + 73T^{2} \) |
| 79 | \( 1 - 8.18T + 79T^{2} \) |
| 83 | \( 1 + 0.679T + 83T^{2} \) |
| 89 | \( 1 + 1.77T + 89T^{2} \) |
| 97 | \( 1 - 5.47T + 97T^{2} \) |
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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
−8.372378959547195672096418232396, −8.123189454543261430375587334552, −6.77024590543191584018466562356, −6.49157100789071853233796073274, −5.83664038597661756947480462394, −4.41698045415064631046974027139, −3.82612910097598934668420692599, −2.60312481903230128713310795477, −2.09429342548682513773491397522, −0.803024804123774092556996641787,
0.803024804123774092556996641787, 2.09429342548682513773491397522, 2.60312481903230128713310795477, 3.82612910097598934668420692599, 4.41698045415064631046974027139, 5.83664038597661756947480462394, 6.49157100789071853233796073274, 6.77024590543191584018466562356, 8.123189454543261430375587334552, 8.372378959547195672096418232396