| L(s) = 1 | − 16·2-s + 256·4-s − 808.·5-s − 4.28e3·7-s − 4.09e3·8-s + 1.29e4·10-s − 9.30e3·11-s − 9.64e4·13-s + 6.85e4·14-s + 6.55e4·16-s − 2.26e5·17-s + 1.30e5·19-s − 2.06e5·20-s + 1.48e5·22-s + 5.65e5·23-s − 1.29e6·25-s + 1.54e6·26-s − 1.09e6·28-s − 7.06e6·29-s − 3.63e6·31-s − 1.04e6·32-s + 3.63e6·34-s + 3.46e6·35-s + 5.99e6·37-s − 2.08e6·38-s + 3.31e6·40-s + 2.57e7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.578·5-s − 0.674·7-s − 0.353·8-s + 0.409·10-s − 0.191·11-s − 0.936·13-s + 0.476·14-s + 0.250·16-s − 0.659·17-s + 0.229·19-s − 0.289·20-s + 0.135·22-s + 0.421·23-s − 0.665·25-s + 0.662·26-s − 0.337·28-s − 1.85·29-s − 0.707·31-s − 0.176·32-s + 0.466·34-s + 0.390·35-s + 0.526·37-s − 0.162·38-s + 0.204·40-s + 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2047399903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2047399903\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 1.30e5T \) |
| good | 5 | \( 1 + 808.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.28e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 9.30e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.64e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.26e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 5.65e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.06e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.63e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.99e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.57e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.04e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.05e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.16e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.37e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.75e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.85e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.57e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.19e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.95e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.51e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.29e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.55e8T + 7.60e17T^{2} \) |
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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
−9.718166239233006310055264618647, −9.271584824869602935812198399219, −8.017956785736747850190248173207, −7.35044618569429656879371948103, −6.41528916983433658051157360242, −5.21727010963727243617350403507, −3.91358256869432692162248014533, −2.85677575901550098943277748983, −1.73009437092723622908276193288, −0.20768964659555789533691625270,
0.20768964659555789533691625270, 1.73009437092723622908276193288, 2.85677575901550098943277748983, 3.91358256869432692162248014533, 5.21727010963727243617350403507, 6.41528916983433658051157360242, 7.35044618569429656879371948103, 8.017956785736747850190248173207, 9.271584824869602935812198399219, 9.718166239233006310055264618647
