| L(s) = 1 | + 1.62·2-s + 1.51·3-s + 0.650·4-s + 1.89·5-s + 2.46·6-s + 7-s − 2.19·8-s − 0.702·9-s + 3.08·10-s − 2.03·11-s + 0.985·12-s − 0.415·13-s + 1.62·14-s + 2.87·15-s − 4.87·16-s − 1.00·17-s − 1.14·18-s − 0.750·19-s + 1.23·20-s + 1.51·21-s − 3.32·22-s + 4.64·23-s − 3.33·24-s − 1.40·25-s − 0.675·26-s − 5.61·27-s + 0.650·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.875·3-s + 0.325·4-s + 0.847·5-s + 1.00·6-s + 0.377·7-s − 0.776·8-s − 0.234·9-s + 0.975·10-s − 0.614·11-s + 0.284·12-s − 0.115·13-s + 0.435·14-s + 0.741·15-s − 1.21·16-s − 0.244·17-s − 0.269·18-s − 0.172·19-s + 0.275·20-s + 0.330·21-s − 0.707·22-s + 0.968·23-s − 0.679·24-s − 0.281·25-s − 0.132·26-s − 1.08·27-s + 0.122·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.814630375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.814630375\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.62T + 2T^{2} \) |
| 3 | \( 1 - 1.51T + 3T^{2} \) |
| 5 | \( 1 - 1.89T + 5T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 + 0.415T + 13T^{2} \) |
| 17 | \( 1 + 1.00T + 17T^{2} \) |
| 19 | \( 1 + 0.750T + 19T^{2} \) |
| 23 | \( 1 - 4.64T + 23T^{2} \) |
| 29 | \( 1 - 6.37T + 29T^{2} \) |
| 31 | \( 1 - 1.05T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 + 8.35T + 41T^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 + 4.85T + 53T^{2} \) |
| 59 | \( 1 + 0.285T + 59T^{2} \) |
| 61 | \( 1 + 8.90T + 61T^{2} \) |
| 67 | \( 1 - 3.33T + 67T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 - 2.72T + 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
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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
−12.01941127671036390876376254051, −10.94059916867066879526433161673, −9.740305323580045169449740344154, −8.894394786790604777587327560432, −8.007774977996019585105171975194, −6.55312224515157899052759050722, −5.52510228592135476436621685445, −4.63957400875972780805929890740, −3.23749117520206814514257971575, −2.30446308741695073081884917481,
2.30446308741695073081884917481, 3.23749117520206814514257971575, 4.63957400875972780805929890740, 5.52510228592135476436621685445, 6.55312224515157899052759050722, 8.007774977996019585105171975194, 8.894394786790604777587327560432, 9.740305323580045169449740344154, 10.94059916867066879526433161673, 12.01941127671036390876376254051
