| L(s) = 1 | + 6.96e3·2-s + 1.50e7·4-s − 4.41e8·5-s + 2.02e10·7-s − 1.29e11·8-s − 3.07e12·10-s + 1.57e13·11-s + 4.98e12·13-s + 1.40e14·14-s − 1.40e15·16-s + 2.34e15·17-s + 1.90e16·19-s − 6.62e15·20-s + 1.10e17·22-s + 6.78e16·23-s − 1.03e17·25-s + 3.47e16·26-s + 3.03e17·28-s + 2.11e18·29-s − 6.62e18·31-s − 5.44e18·32-s + 1.63e19·34-s − 8.92e18·35-s + 1.02e18·37-s + 1.32e20·38-s + 5.70e19·40-s + 1.74e20·41-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 0.447·4-s − 0.808·5-s + 0.552·7-s − 0.664·8-s − 0.972·10-s + 1.51·11-s + 0.0593·13-s + 0.664·14-s − 1.24·16-s + 0.976·17-s + 1.97·19-s − 0.361·20-s + 1.82·22-s + 0.645·23-s − 0.346·25-s + 0.0714·26-s + 0.247·28-s + 1.10·29-s − 1.51·31-s − 0.835·32-s + 1.17·34-s − 0.446·35-s + 0.0256·37-s + 2.37·38-s + 0.537·40-s + 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(3.619036496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.619036496\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 6.96e3T + 3.35e7T^{2} \) |
| 5 | \( 1 + 4.41e8T + 2.98e17T^{2} \) |
| 7 | \( 1 - 2.02e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.57e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 4.98e12T + 7.05e27T^{2} \) |
| 17 | \( 1 - 2.34e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.90e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 6.78e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 2.11e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 6.62e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 1.02e18T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.74e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 3.26e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 4.36e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 4.14e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.51e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 4.63e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 1.21e23T + 4.48e45T^{2} \) |
| 71 | \( 1 + 1.87e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 4.86e22T + 3.82e46T^{2} \) |
| 79 | \( 1 - 4.43e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.52e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.97e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 3.28e24T + 4.66e49T^{2} \) |
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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
−14.77093588214545654446446926446, −13.96813089087798739572828127917, −12.23201780983411527755688966335, −11.49606583997323559537793072120, −9.180741226354052481359183503806, −7.38773330123260946890575673582, −5.64783095105107578333774861197, −4.26422460989554818025128684485, −3.25193321240808968731774238468, −1.04707398580710961070066808697,
1.04707398580710961070066808697, 3.25193321240808968731774238468, 4.26422460989554818025128684485, 5.64783095105107578333774861197, 7.38773330123260946890575673582, 9.180741226354052481359183503806, 11.49606583997323559537793072120, 12.23201780983411527755688966335, 13.96813089087798739572828127917, 14.77093588214545654446446926446
