| L(s) = 1 | + 5.31e5·3-s + 1.03e9·5-s + 3.32e10·7-s + 2.82e11·9-s + 7.60e12·11-s + 5.42e13·13-s + 5.50e14·15-s + 2.53e15·17-s + 1.35e15·19-s + 1.76e16·21-s − 1.50e17·23-s + 7.74e17·25-s + 1.50e17·27-s − 2.24e18·29-s + 6.09e18·31-s + 4.04e18·33-s + 3.44e19·35-s − 2.76e19·37-s + 2.88e19·39-s + 2.52e20·41-s + 2.51e20·43-s + 2.92e20·45-s − 4.65e19·47-s − 2.37e20·49-s + 1.34e21·51-s + 3.32e21·53-s + 7.87e21·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.89·5-s + 0.907·7-s + 0.333·9-s + 0.730·11-s + 0.646·13-s + 1.09·15-s + 1.05·17-s + 0.140·19-s + 0.523·21-s − 1.43·23-s + 2.59·25-s + 0.192·27-s − 1.18·29-s + 1.39·31-s + 0.421·33-s + 1.72·35-s − 0.691·37-s + 0.373·39-s + 1.74·41-s + 0.958·43-s + 0.632·45-s − 0.0584·47-s − 0.177·49-s + 0.609·51-s + 0.930·53-s + 1.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(6.172280089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.172280089\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.31e5T \) |
| good | 5 | \( 1 - 1.03e9T + 2.98e17T^{2} \) |
| 7 | \( 1 - 3.32e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 7.60e12T + 1.08e26T^{2} \) |
| 13 | \( 1 - 5.42e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 2.53e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.35e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 1.50e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 2.24e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 6.09e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 2.76e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 2.52e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 2.51e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 4.65e19T + 6.34e41T^{2} \) |
| 53 | \( 1 - 3.32e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.91e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 1.45e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 3.78e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 1.33e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 2.87e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 2.03e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 4.22e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 3.33e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 2.95e24T + 4.66e49T^{2} \) |
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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
−10.67321492280327807317513138149, −9.724248554838000927289918554401, −8.916600249249991935211121144210, −7.73605066922338466603318179515, −6.25817748289197178321049141222, −5.51804234032833282296454347405, −4.16262228776376066491782153026, −2.73754567272669509369242514065, −1.70340678770109061230423241319, −1.18891002909417336974421825005,
1.18891002909417336974421825005, 1.70340678770109061230423241319, 2.73754567272669509369242514065, 4.16262228776376066491782153026, 5.51804234032833282296454347405, 6.25817748289197178321049141222, 7.73605066922338466603318179515, 8.916600249249991935211121144210, 9.724248554838000927289918554401, 10.67321492280327807317513138149
