L(s) = 1 | − 941. i·5-s + 907.·7-s − 2.49e4i·11-s − 5.39e4·13-s − 1.26e5i·17-s + 2.19e3·19-s + 4.99e5i·23-s − 4.95e5·25-s − 4.71e5i·29-s − 1.20e6·31-s − 8.54e5i·35-s − 9.05e4·37-s − 5.01e6i·41-s + 1.25e6·43-s + 4.26e6i·47-s + ⋯ |
L(s) = 1 | − 1.50i·5-s + 0.377·7-s − 1.70i·11-s − 1.88·13-s − 1.51i·17-s + 0.0168·19-s + 1.78i·23-s − 1.26·25-s − 0.666i·29-s − 1.30·31-s − 0.569i·35-s − 0.0483·37-s − 1.77i·41-s + 0.367·43-s + 0.873i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7319113571\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7319113571\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 907.T \) |
good | 5 | \( 1 + 941. iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 2.49e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 5.39e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.26e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 2.19e3T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.99e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.71e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.20e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 9.05e4T + 3.51e12T^{2} \) |
| 41 | \( 1 + 5.01e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.25e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 4.26e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 2.44e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 3.63e5iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.52e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.86e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 6.10e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 4.56e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 6.35e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.96e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 6.47e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.05e8T + 7.83e15T^{2} \) |
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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
−9.642439115835983813527928931499, −9.129587600079645385863283933492, −8.085770249641542958165560840077, −7.23966548944690581783587801605, −5.40379355756178256480231919083, −5.20267720689926084352089614164, −3.79342077490434733317264271065, −2.35129462199676547056356385662, −0.940079226487895690371920762823, −0.17491507130467527216832065476,
1.93925975751902134222226731424, 2.59711563311538070964753679107, 4.05531241711048958968689057058, 5.09450715579257610846750241607, 6.63138660702401602405265711805, 7.14376729188096505040618348231, 8.105785640978262691118439413512, 9.659221876159246372944660262921, 10.28482694919429912506257048224, 11.02507083037095731552671800850