| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 5·5-s + 6·6-s + 28.5·7-s + 8·8-s + 9·9-s + 10·10-s − 46.5·11-s + 12·12-s + 25.2·13-s + 57.1·14-s + 15·15-s + 16·16-s − 6.54·17-s + 18·18-s + 57.9·19-s + 20·20-s + 85.7·21-s − 93.0·22-s + 52.0·23-s + 24·24-s + 25·25-s + 50.4·26-s + 27·27-s + 114.·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.54·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 1.27·11-s + 0.288·12-s + 0.538·13-s + 1.09·14-s + 0.258·15-s + 0.250·16-s − 0.0934·17-s + 0.235·18-s + 0.700·19-s + 0.223·20-s + 0.890·21-s − 0.901·22-s + 0.471·23-s + 0.204·24-s + 0.200·25-s + 0.380·26-s + 0.192·27-s + 0.771·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.485915770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.485915770\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 31 | \( 1 - 31T \) |
| good | 7 | \( 1 - 28.5T + 343T^{2} \) |
| 11 | \( 1 + 46.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 25.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 57.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 52.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 80.7T + 2.43e4T^{2} \) |
| 37 | \( 1 - 164.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 279.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 401.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 170.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 323.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 18.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 623.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 219.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 78.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 464.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 325.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 127.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 672.T + 9.12e5T^{2} \) |
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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
−9.761309947226699116335455446364, −8.631038805055097252612902648829, −7.964279806671158532139792914592, −7.28531267993273735301451450293, −6.03153020553103087236337278514, −5.10760307471864125598982598456, −4.54563199665211702281510160475, −3.20115767209935280866654873137, −2.26530775645331728865932565768, −1.25061176733427370639875388341,
1.25061176733427370639875388341, 2.26530775645331728865932565768, 3.20115767209935280866654873137, 4.54563199665211702281510160475, 5.10760307471864125598982598456, 6.03153020553103087236337278514, 7.28531267993273735301451450293, 7.964279806671158532139792914592, 8.631038805055097252612902648829, 9.761309947226699116335455446364
