| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 8.62·5-s − 6·6-s + 7.84·7-s − 8·8-s + 9·9-s − 17.2·10-s − 3.30·11-s + 12·12-s + 18.7·13-s − 15.6·14-s + 25.8·15-s + 16·16-s + 97.3·17-s − 18·18-s + 41.1·19-s + 34.5·20-s + 23.5·21-s + 6.61·22-s − 21.6·23-s − 24·24-s − 50.5·25-s − 37.4·26-s + 27·27-s + 31.3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.771·5-s − 0.408·6-s + 0.423·7-s − 0.353·8-s + 0.333·9-s − 0.545·10-s − 0.0906·11-s + 0.288·12-s + 0.399·13-s − 0.299·14-s + 0.445·15-s + 0.250·16-s + 1.38·17-s − 0.235·18-s + 0.496·19-s + 0.385·20-s + 0.244·21-s + 0.0640·22-s − 0.196·23-s − 0.204·24-s − 0.404·25-s − 0.282·26-s + 0.192·27-s + 0.211·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.141398133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141398133\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 59 | \( 1 - 59T \) |
| good | 5 | \( 1 - 8.62T + 125T^{2} \) |
| 7 | \( 1 - 7.84T + 343T^{2} \) |
| 11 | \( 1 + 3.30T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 97.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 41.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 21.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 287.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 351.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 84.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 76.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 168.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 133.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 246.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 396.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 172.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 959.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 78.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 318.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 875.T + 9.12e5T^{2} \) |
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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
−10.86669037080866308017220482400, −9.773896189546939139374869995545, −9.431892669971900223319947547336, −8.151793088620687335219406175131, −7.62506753341089263452938554696, −6.28101340654121667295258180200, −5.31255964341133141278929096191, −3.65750032727101024722091834007, −2.32389002063092961637461371904, −1.16470883402328882636104057780,
1.16470883402328882636104057780, 2.32389002063092961637461371904, 3.65750032727101024722091834007, 5.31255964341133141278929096191, 6.28101340654121667295258180200, 7.62506753341089263452938554696, 8.151793088620687335219406175131, 9.431892669971900223319947547336, 9.773896189546939139374869995545, 10.86669037080866308017220482400
