| L(s) = 1 | + 1.73·2-s − 3-s + 0.999·4-s − 2.73·5-s − 1.73·6-s − 2·7-s − 1.73·8-s + 9-s − 4.73·10-s − 11-s − 0.999·12-s − 13-s − 3.46·14-s + 2.73·15-s − 5·16-s + 3.26·17-s + 1.73·18-s − 7.46·19-s − 2.73·20-s + 2·21-s − 1.73·22-s − 2·23-s + 1.73·24-s + 2.46·25-s − 1.73·26-s − 27-s − 1.99·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.577·3-s + 0.499·4-s − 1.22·5-s − 0.707·6-s − 0.755·7-s − 0.612·8-s + 0.333·9-s − 1.49·10-s − 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.925·14-s + 0.705·15-s − 1.25·16-s + 0.792·17-s + 0.408·18-s − 1.71·19-s − 0.610·20-s + 0.436·21-s − 0.369·22-s − 0.417·23-s + 0.353·24-s + 0.492·25-s − 0.339·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 - 5.46T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 6.53T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 1.80T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 1.66T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−10.98233132089374916908110440536, −10.09122356878340641242633577971, −8.810786877525986769989149208569, −7.73888656519421108670134539450, −6.62780667496321996725889291267, −5.85709950063880528463459733171, −4.64213184103502313720127885293, −3.97710819556428815156256206048, −2.87032313039725980496591990680, 0,
2.87032313039725980496591990680, 3.97710819556428815156256206048, 4.64213184103502313720127885293, 5.85709950063880528463459733171, 6.62780667496321996725889291267, 7.73888656519421108670134539450, 8.810786877525986769989149208569, 10.09122356878340641242633577971, 10.98233132089374916908110440536
