L(s) = 1 | − 2-s + 0.585·3-s + 4-s + 5-s − 0.585·6-s − 8-s − 2.65·9-s − 10-s + 4.82·11-s + 0.585·12-s + 0.828·13-s + 0.585·15-s + 16-s + 5.41·17-s + 2.65·18-s + 3.41·19-s + 20-s − 4.82·22-s − 6.82·23-s − 0.585·24-s + 25-s − 0.828·26-s − 3.31·27-s + 0.828·29-s − 0.585·30-s + 2.82·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.338·3-s + 0.5·4-s + 0.447·5-s − 0.239·6-s − 0.353·8-s − 0.885·9-s − 0.316·10-s + 1.45·11-s + 0.169·12-s + 0.229·13-s + 0.151·15-s + 0.250·16-s + 1.31·17-s + 0.626·18-s + 0.783·19-s + 0.223·20-s − 1.02·22-s − 1.42·23-s − 0.119·24-s + 0.200·25-s − 0.162·26-s − 0.637·27-s + 0.153·29-s − 0.106·30-s + 0.508·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.298609393\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.298609393\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 0.585T + 3T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 9.65T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 6.58T + 73T^{2} \) |
| 79 | \( 1 + 1.17T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−10.88459580007985610221619904271, −9.749815474932624267620391551470, −9.318640555922559278614043276088, −8.342847444009443390803294271408, −7.56099190688200919690633753526, −6.30460120710322372141500888161, −5.65977706098032818667730898633, −3.93324940979279260936738912402, −2.76040274213252714370841685164, −1.28269316591121626929788201140,
1.28269316591121626929788201140, 2.76040274213252714370841685164, 3.93324940979279260936738912402, 5.65977706098032818667730898633, 6.30460120710322372141500888161, 7.56099190688200919690633753526, 8.342847444009443390803294271408, 9.318640555922559278614043276088, 9.749815474932624267620391551470, 10.88459580007985610221619904271