| L(s) = 1 | + 6.08·2-s + 9·3-s + 20.9·4-s − 12.1·5-s + 54.7·6-s + 50·7-s + 30.4·8-s + 81·9-s − 73.9·10-s + 188.·12-s + 304.·14-s − 109.·15-s − 151.·16-s + 279.·17-s + 492.·18-s − 255.·20-s + 450·21-s − 888.·23-s + 273.·24-s − 477·25-s + 729·27-s + 1.04e3·28-s − 1.18e3·29-s − 665.·30-s − 1.40e3·32-s + 1.70e3·34-s − 608.·35-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 3-s + 1.31·4-s − 0.486·5-s + 1.52·6-s + 1.02·7-s + 0.475·8-s + 9-s − 0.739·10-s + 1.31·12-s + 1.55·14-s − 0.486·15-s − 0.589·16-s + 0.968·17-s + 1.52·18-s − 0.638·20-s + 1.02·21-s − 1.67·23-s + 0.475·24-s − 0.763·25-s + 0.999·27-s + 1.33·28-s − 1.40·29-s − 0.739·30-s − 1.37·32-s + 1.47·34-s − 0.496·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.184398310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.184398310\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 37 | \( 1 - 1.36e3T \) |
| good | 2 | \( 1 - 6.08T + 16T^{2} \) |
| 5 | \( 1 + 12.1T + 625T^{2} \) |
| 7 | \( 1 - 50T + 2.40e3T^{2} \) |
| 11 | \( 1 - 1.46e4T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 17 | \( 1 - 279.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 1.30e5T^{2} \) |
| 23 | \( 1 + 888.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.18e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.73e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 1.38e7T^{2} \) |
| 67 | \( 1 - 8.93e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.05e4T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.89e7T^{2} \) |
| 83 | \( 1 - 4.74e7T^{2} \) |
| 89 | \( 1 + 888.T + 6.27e7T^{2} \) |
| 97 | \( 1 - 8.85e7T^{2} \) |
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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
−13.13916771546090101766512976940, −12.14326001967644532507276886921, −11.29880783532070366266724824315, −9.765420254346972005104871733832, −8.255954817774962630960375153879, −7.42525042718145371270418730942, −5.76158093520027846774528211910, −4.41955339471520216680867924491, −3.56008674067624716916950586329, −2.02593834258199517333018057904,
2.02593834258199517333018057904, 3.56008674067624716916950586329, 4.41955339471520216680867924491, 5.76158093520027846774528211910, 7.42525042718145371270418730942, 8.255954817774962630960375153879, 9.765420254346972005104871733832, 11.29880783532070366266724824315, 12.14326001967644532507276886921, 13.13916771546090101766512976940
