| L(s) = 1 | − 16·2-s + 123.·3-s + 256·4-s + 1.18e3·5-s − 1.97e3·6-s + 1.17e3·7-s − 4.09e3·8-s − 4.38e3·9-s − 1.89e4·10-s + 1.46e4·11-s + 3.16e4·12-s + 8.24e4·13-s − 1.87e4·14-s + 1.46e5·15-s + 6.55e4·16-s + 4.40e5·17-s + 7.02e4·18-s + 8.04e5·19-s + 3.03e5·20-s + 1.45e5·21-s − 2.34e5·22-s − 2.28e5·23-s − 5.06e5·24-s − 5.47e5·25-s − 1.31e6·26-s − 2.97e6·27-s + 3.00e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.881·3-s + 0.5·4-s + 0.848·5-s − 0.623·6-s + 0.184·7-s − 0.353·8-s − 0.222·9-s − 0.599·10-s + 0.301·11-s + 0.440·12-s + 0.800·13-s − 0.130·14-s + 0.747·15-s + 0.250·16-s + 1.28·17-s + 0.157·18-s + 1.41·19-s + 0.424·20-s + 0.163·21-s − 0.213·22-s − 0.170·23-s − 0.311·24-s − 0.280·25-s − 0.566·26-s − 1.07·27-s + 0.0924·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.040013188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.040013188\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 11 | \( 1 - 1.46e4T \) |
| good | 3 | \( 1 - 123.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.18e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.17e3T + 4.03e7T^{2} \) |
| 13 | \( 1 - 8.24e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.40e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.28e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.56e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.68e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.30e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.81e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.85e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.97e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.31e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.12e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.27e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.55e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.96e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.00e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.29e9T + 7.60e17T^{2} \) |
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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
−15.98862612993314410639477841416, −14.46831021001282356892790031732, −13.63124310535358002612806001329, −11.75975307575129699898920843945, −10.07948358614585014368763106035, −9.020640629053368125138312110931, −7.77038861335200721854753370759, −5.85929282597650854684097094746, −3.10734395935336692249976399591, −1.41581467294580684583261045009,
1.41581467294580684583261045009, 3.10734395935336692249976399591, 5.85929282597650854684097094746, 7.77038861335200721854753370759, 9.020640629053368125138312110931, 10.07948358614585014368763106035, 11.75975307575129699898920843945, 13.63124310535358002612806001329, 14.46831021001282356892790031732, 15.98862612993314410639477841416
