L(s) = 1 | + 2.29·2-s − 3-s + 3.26·4-s − 4.16·5-s − 2.29·6-s − 2.32·7-s + 2.89·8-s + 9-s − 9.55·10-s − 1.76·11-s − 3.26·12-s − 13-s − 5.34·14-s + 4.16·15-s + 0.125·16-s − 0.447·17-s + 2.29·18-s − 1.95·19-s − 13.5·20-s + 2.32·21-s − 4.05·22-s + 1.95·23-s − 2.89·24-s + 12.3·25-s − 2.29·26-s − 27-s − 7.60·28-s + ⋯ |
L(s) = 1 | + 1.62·2-s − 0.577·3-s + 1.63·4-s − 1.86·5-s − 0.936·6-s − 0.880·7-s + 1.02·8-s + 0.333·9-s − 3.02·10-s − 0.533·11-s − 0.942·12-s − 0.277·13-s − 1.42·14-s + 1.07·15-s + 0.0312·16-s − 0.108·17-s + 0.540·18-s − 0.447·19-s − 3.03·20-s + 0.508·21-s − 0.864·22-s + 0.408·23-s − 0.591·24-s + 2.46·25-s − 0.449·26-s − 0.192·27-s − 1.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.764087648\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.764087648\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 2.29T + 2T^{2} \) |
| 5 | \( 1 + 4.16T + 5T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 17 | \( 1 + 0.447T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 - 6.09T + 29T^{2} \) |
| 31 | \( 1 - 3.55T + 31T^{2} \) |
| 37 | \( 1 + 5.25T + 37T^{2} \) |
| 41 | \( 1 - 5.41T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 - 2.22T + 53T^{2} \) |
| 59 | \( 1 - 0.171T + 59T^{2} \) |
| 61 | \( 1 - 1.59T + 61T^{2} \) |
| 67 | \( 1 + 8.82T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 4.04T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.194513361714958019264432872543, −7.32259759216907057093316669462, −6.87313614166029679851799596379, −6.13456439913708343370424594017, −5.27898264324045581088554467595, −4.48189803389172329053248231320, −4.09077373227580255381515638048, −3.22180856688843490534848481087, −2.60012491628941254620052643305, −0.58220990220629931603907501038,
0.58220990220629931603907501038, 2.60012491628941254620052643305, 3.22180856688843490534848481087, 4.09077373227580255381515638048, 4.48189803389172329053248231320, 5.27898264324045581088554467595, 6.13456439913708343370424594017, 6.87313614166029679851799596379, 7.32259759216907057093316669462, 8.194513361714958019264432872543