| L(s) = 1 | + 41.8·2-s − 0.232·3-s + 1.23e3·4-s + 1.79e3·5-s − 9.71·6-s + 3.02e4·8-s − 1.96e4·9-s + 7.49e4·10-s + 1.74e4·11-s − 287.·12-s + 1.22e5·13-s − 416.·15-s + 6.31e5·16-s − 3.31e5·17-s − 8.22e5·18-s − 7.61e5·19-s + 2.21e6·20-s + 7.27e5·22-s + 1.23e6·23-s − 7.02e3·24-s + 1.25e6·25-s + 5.12e6·26-s + 9.14e3·27-s + 6.34e5·29-s − 1.74e4·30-s + 5.38e6·31-s + 1.09e7·32-s + ⋯ |
| L(s) = 1 | + 1.84·2-s − 0.00165·3-s + 2.41·4-s + 1.28·5-s − 0.00305·6-s + 2.61·8-s − 0.999·9-s + 2.36·10-s + 0.358·11-s − 0.00399·12-s + 1.18·13-s − 0.00212·15-s + 2.40·16-s − 0.963·17-s − 1.84·18-s − 1.34·19-s + 3.09·20-s + 0.662·22-s + 0.918·23-s − 0.00432·24-s + 0.643·25-s + 2.19·26-s + 0.00331·27-s + 0.166·29-s − 0.00392·30-s + 1.04·31-s + 1.84·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(7.157278189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.157278189\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 41.8T + 512T^{2} \) |
| 3 | \( 1 + 0.232T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.79e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 1.74e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.22e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.31e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.61e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.23e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.34e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.03e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.37e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.97e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.44e6T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.19e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 5.58e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.54e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.51e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.34e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.51e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.42e9T + 7.60e17T^{2} \) |
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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
−13.59270988755679941921884573219, −12.97775085981026659509640557123, −11.54264747497163780752124895529, −10.63591792291448371981357466465, −8.759747873440187258200565025844, −6.49272240549395913862677195741, −5.96639948110595423697251286766, −4.61715652542764830031876041140, −3.04013122834108640926785942800, −1.82067337280000974999082691148,
1.82067337280000974999082691148, 3.04013122834108640926785942800, 4.61715652542764830031876041140, 5.96639948110595423697251286766, 6.49272240549395913862677195741, 8.759747873440187258200565025844, 10.63591792291448371981357466465, 11.54264747497163780752124895529, 12.97775085981026659509640557123, 13.59270988755679941921884573219
