| L(s) = 1 | − 1.01e10·3-s + 6.19e14·5-s − 2.58e18·7-s − 2.25e20·9-s − 2.74e22·11-s + 6.62e23·13-s − 6.29e24·15-s + 5.82e25·17-s + 1.98e27·19-s + 2.62e28·21-s + 2.61e29·23-s − 7.52e29·25-s + 5.61e30·27-s + 2.27e31·29-s + 1.73e32·31-s + 2.78e32·33-s − 1.60e33·35-s + 5.95e32·37-s − 6.72e33·39-s − 5.56e34·41-s − 1.13e35·43-s − 1.39e35·45-s − 2.48e35·47-s + 4.50e36·49-s − 5.90e35·51-s + 3.93e36·53-s − 1.70e37·55-s + ⋯ |
| L(s) = 1 | − 0.560·3-s + 0.581·5-s − 1.75·7-s − 0.686·9-s − 1.11·11-s + 0.743·13-s − 0.325·15-s + 0.204·17-s + 0.636·19-s + 0.980·21-s + 1.37·23-s − 0.662·25-s + 0.944·27-s + 0.824·29-s + 1.49·31-s + 0.626·33-s − 1.01·35-s + 0.114·37-s − 0.416·39-s − 1.17·41-s − 0.865·43-s − 0.398·45-s − 0.279·47-s + 2.06·49-s − 0.114·51-s + 0.333·53-s − 0.649·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 1.01e10T + 3.28e20T^{2} \) |
| 5 | \( 1 - 6.19e14T + 1.13e30T^{2} \) |
| 7 | \( 1 + 2.58e18T + 2.18e36T^{2} \) |
| 11 | \( 1 + 2.74e22T + 6.02e44T^{2} \) |
| 13 | \( 1 - 6.62e23T + 7.93e47T^{2} \) |
| 17 | \( 1 - 5.82e25T + 8.11e52T^{2} \) |
| 19 | \( 1 - 1.98e27T + 9.69e54T^{2} \) |
| 23 | \( 1 - 2.61e29T + 3.58e58T^{2} \) |
| 29 | \( 1 - 2.27e31T + 7.64e62T^{2} \) |
| 31 | \( 1 - 1.73e32T + 1.34e64T^{2} \) |
| 37 | \( 1 - 5.95e32T + 2.70e67T^{2} \) |
| 41 | \( 1 + 5.56e34T + 2.23e69T^{2} \) |
| 43 | \( 1 + 1.13e35T + 1.73e70T^{2} \) |
| 47 | \( 1 + 2.48e35T + 7.94e71T^{2} \) |
| 53 | \( 1 - 3.93e36T + 1.39e74T^{2} \) |
| 59 | \( 1 + 6.06e37T + 1.40e76T^{2} \) |
| 61 | \( 1 - 3.31e38T + 5.87e76T^{2} \) |
| 67 | \( 1 + 2.46e38T + 3.32e78T^{2} \) |
| 71 | \( 1 - 5.22e39T + 4.01e79T^{2} \) |
| 73 | \( 1 + 1.32e40T + 1.32e80T^{2} \) |
| 79 | \( 1 - 1.99e40T + 3.96e81T^{2} \) |
| 83 | \( 1 - 5.56e39T + 3.31e82T^{2} \) |
| 89 | \( 1 + 2.64e41T + 6.66e83T^{2} \) |
| 97 | \( 1 - 3.46e42T + 2.69e85T^{2} \) |
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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
−10.49512312760056659698817132279, −9.699387355000175627031550736889, −8.433769324198956889791718053261, −6.79113116879165612274287575969, −5.99962449566981373697633648621, −5.11058865712505620351615083938, −3.31527973296784594901915008109, −2.65999247952819467966859925671, −0.934894964646956320275881702013, 0,
0.934894964646956320275881702013, 2.65999247952819467966859925671, 3.31527973296784594901915008109, 5.11058865712505620351615083938, 5.99962449566981373697633648621, 6.79113116879165612274287575969, 8.433769324198956889791718053261, 9.699387355000175627031550736889, 10.49512312760056659698817132279
