| L(s) = 1 | + 16·2-s + 266.·3-s + 256·4-s + 745.·5-s + 4.27e3·6-s − 3.11e3·7-s + 4.09e3·8-s + 5.15e4·9-s + 1.19e4·10-s − 7.10e3·11-s + 6.83e4·12-s − 6.93e3·13-s − 4.98e4·14-s + 1.99e5·15-s + 6.55e4·16-s − 3.27e5·17-s + 8.25e5·18-s − 1.30e5·19-s + 1.90e5·20-s − 8.31e5·21-s − 1.13e5·22-s − 9.55e5·23-s + 1.09e6·24-s − 1.39e6·25-s − 1.10e5·26-s + 8.52e6·27-s − 7.97e5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.90·3-s + 0.5·4-s + 0.533·5-s + 1.34·6-s − 0.490·7-s + 0.353·8-s + 2.62·9-s + 0.377·10-s − 0.146·11-s + 0.951·12-s − 0.0673·13-s − 0.346·14-s + 1.01·15-s + 0.250·16-s − 0.950·17-s + 1.85·18-s − 0.229·19-s + 0.266·20-s − 0.933·21-s − 0.103·22-s − 0.711·23-s + 0.672·24-s − 0.715·25-s − 0.0476·26-s + 3.08·27-s − 0.245·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.569372678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.569372678\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 3 | \( 1 - 266.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 745.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.11e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.10e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.93e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.27e5T + 1.18e11T^{2} \) |
| 23 | \( 1 + 9.55e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.95e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.62e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.94e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.62e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.75e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.09e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.17e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.66e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.58e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.80e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.20e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.99e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.03e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.10e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.04e8T + 7.60e17T^{2} \) |
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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
−14.00983030709919458467065728796, −13.54228610132954016835052938274, −12.44792246869048815421292260987, −10.30632513617105200964319312286, −9.219743575779174238093632426688, −7.951545027585531099812787873668, −6.52341566069368986167758514889, −4.34721761367068607904431943602, −2.99996472236985578391572855424, −1.92609214215660153127208254442,
1.92609214215660153127208254442, 2.99996472236985578391572855424, 4.34721761367068607904431943602, 6.52341566069368986167758514889, 7.951545027585531099812787873668, 9.219743575779174238093632426688, 10.30632513617105200964319312286, 12.44792246869048815421292260987, 13.54228610132954016835052938274, 14.00983030709919458467065728796
