| L(s) = 1 | − 105.·2-s + 6.56e3·3-s − 1.19e5·4-s − 3.90e5·5-s − 6.92e5·6-s − 2.39e7·7-s + 2.64e7·8-s + 4.30e7·9-s + 4.12e7·10-s − 5.09e8·11-s − 7.86e8·12-s + 4.71e9·13-s + 2.52e9·14-s − 2.56e9·15-s + 1.29e10·16-s − 4.44e10·17-s − 4.54e9·18-s − 4.25e10·19-s + 4.68e10·20-s − 1.57e11·21-s + 5.37e10·22-s + 4.70e11·23-s + 1.73e11·24-s + 1.52e11·25-s − 4.97e11·26-s + 2.82e11·27-s + 2.87e12·28-s + ⋯ |
| L(s) = 1 | − 0.291·2-s + 0.577·3-s − 0.915·4-s − 0.447·5-s − 0.168·6-s − 1.56·7-s + 0.558·8-s + 0.333·9-s + 0.130·10-s − 0.716·11-s − 0.528·12-s + 1.60·13-s + 0.457·14-s − 0.258·15-s + 0.752·16-s − 1.54·17-s − 0.0971·18-s − 0.574·19-s + 0.409·20-s − 0.906·21-s + 0.208·22-s + 1.25·23-s + 0.322·24-s + 0.200·25-s − 0.467·26-s + 0.192·27-s + 1.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.019621990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019621990\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 2 | \( 1 + 105.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 2.39e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 5.09e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 4.71e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 4.44e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 4.25e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 4.70e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 3.15e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 3.46e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 3.18e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 8.13e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.24e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + 7.05e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.51e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 6.39e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 4.96e13T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.75e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 2.50e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 8.85e14T + 4.74e31T^{2} \) |
| 79 | \( 1 + 6.85e13T + 1.81e32T^{2} \) |
| 83 | \( 1 - 3.38e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.37e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 6.48e16T + 5.95e33T^{2} \) |
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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
−15.34196715052449131631863753528, −13.42843431386811350689960734378, −13.00161100844048861381802560806, −10.67799851501047036353368478062, −9.273147945471900130708433227528, −8.324007795617635274848074870538, −6.49096354569961226466899759402, −4.30866501748368835500625759353, −3.01222416098395716802918697862, −0.65471667732610118785935976027,
0.65471667732610118785935976027, 3.01222416098395716802918697862, 4.30866501748368835500625759353, 6.49096354569961226466899759402, 8.324007795617635274848074870538, 9.273147945471900130708433227528, 10.67799851501047036353368478062, 13.00161100844048861381802560806, 13.42843431386811350689960734378, 15.34196715052449131631863753528
