| L(s) = 1 | + 70.4·3-s + 164.·5-s + 343·7-s + 2.77e3·9-s + 4.56e3·11-s − 2.03e3·13-s + 1.15e4·15-s − 1.73e4·17-s + 5.88e3·19-s + 2.41e4·21-s + 4.11e4·23-s − 5.11e4·25-s + 4.13e4·27-s + 1.41e5·29-s + 1.70e5·31-s + 3.21e5·33-s + 5.63e4·35-s + 2.25e5·37-s − 1.43e5·39-s + 6.94e5·41-s − 7.49e5·43-s + 4.55e5·45-s − 1.34e6·47-s + 1.17e5·49-s − 1.22e6·51-s − 1.79e6·53-s + 7.50e5·55-s + ⋯ |
| L(s) = 1 | + 1.50·3-s + 0.588·5-s + 0.377·7-s + 1.26·9-s + 1.03·11-s − 0.256·13-s + 0.885·15-s − 0.856·17-s + 0.196·19-s + 0.569·21-s + 0.705·23-s − 0.654·25-s + 0.404·27-s + 1.07·29-s + 1.02·31-s + 1.55·33-s + 0.222·35-s + 0.730·37-s − 0.386·39-s + 1.57·41-s − 1.43·43-s + 0.745·45-s − 1.89·47-s + 0.142·49-s − 1.28·51-s − 1.65·53-s + 0.608·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.602811516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.602811516\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 - 70.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 164.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.56e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.03e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.73e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.88e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.11e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.70e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.25e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.94e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.49e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.34e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.79e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.15e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 9.79e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.25e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.24e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.29e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.19e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.78e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.74e6T + 8.07e13T^{2} \) |
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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
−13.96923482332690022078169846795, −13.03059363326415130965369923361, −11.50183878244099747206147746655, −9.858491338507152073165805290296, −9.020172016437765966174965769296, −7.974241622103031375405165734530, −6.51138262239840106887462084525, −4.46804992645736447182567342138, −2.90497607922599769238393961142, −1.58705611455035689503648348742,
1.58705611455035689503648348742, 2.90497607922599769238393961142, 4.46804992645736447182567342138, 6.51138262239840106887462084525, 7.974241622103031375405165734530, 9.020172016437765966174965769296, 9.858491338507152073165805290296, 11.50183878244099747206147746655, 13.03059363326415130965369923361, 13.96923482332690022078169846795
