L(s) = 1 | + 2.67·2-s + 3-s + 5.15·4-s − 0.481·5-s + 2.67·6-s − 4.15·7-s + 8.44·8-s + 9-s − 1.28·10-s − 11-s + 5.15·12-s − 13-s − 11.1·14-s − 0.481·15-s + 12.2·16-s − 1.67·17-s + 2.67·18-s + 4.15·19-s − 2.48·20-s − 4.15·21-s − 2.67·22-s − 5.50·23-s + 8.44·24-s − 4.76·25-s − 2.67·26-s + 27-s − 21.4·28-s + ⋯ |
L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.57·4-s − 0.215·5-s + 1.09·6-s − 1.57·7-s + 2.98·8-s + 0.333·9-s − 0.407·10-s − 0.301·11-s + 1.48·12-s − 0.277·13-s − 2.97·14-s − 0.124·15-s + 3.06·16-s − 0.406·17-s + 0.630·18-s + 0.953·19-s − 0.554·20-s − 0.906·21-s − 0.570·22-s − 1.14·23-s + 1.72·24-s − 0.953·25-s − 0.524·26-s + 0.192·27-s − 4.05·28-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
\(L(1)\) |
\(\approx\) |
\(4.119410879\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.119410879\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 + 0.481T + 5T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 + 5.50T + 23T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 - 2.15T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 - 8.57T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 7.66T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 9.47T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 - 8.96T + 83T^{2} \) |
| 89 | \( 1 - 2.48T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−11.67551806700917807474034922967, −10.34621603135489014254872045491, −9.669412849485241661081748318746, −8.088314656115432470468388245891, −7.02457230757647359030767263560, −6.33825554731416592035776507122, −5.30762276865224137516316276676, −4.06050500485667345506197370127, −3.30994060828330222802108136608, −2.37502782587245148335031686945,
2.37502782587245148335031686945, 3.30994060828330222802108136608, 4.06050500485667345506197370127, 5.30762276865224137516316276676, 6.33825554731416592035776507122, 7.02457230757647359030767263560, 8.088314656115432470468388245891, 9.669412849485241661081748318746, 10.34621603135489014254872045491, 11.67551806700917807474034922967