L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + (2 + 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 7·13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (−0.5 + 0.866i)19-s − 0.999·20-s − 0.999·22-s + (0.5 − 0.866i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + (0.755 + 0.654i)7-s − 0.353·8-s + (−0.158 − 0.273i)10-s + (−0.150 − 0.261i)11-s + 1.94·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (−0.114 + 0.198i)19-s − 0.223·20-s − 0.213·22-s + (0.104 − 0.180i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67027 - 1.11103i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67027 - 1.11103i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 11 | \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 7T + 13T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−10.69801583489640963834149074405, −9.572632805190784027739284739037, −8.666585487367050046528870742845, −8.202568889945152015460403992989, −6.59300376078337202567710398881, −5.69958069259223045967488953392, −4.85809735754376959928951563893, −3.77626609043496592640883211206, −2.47701075094169862455421068176, −1.21355741267945689276250842244,
1.56088455202916376653480763358, 3.33724876461643906643390240718, 4.26974582223375506078480735217, 5.31175734659128630474313456349, 6.41903359710852254102492728028, 6.97359734140330362362674046285, 8.312231360735852369355269882641, 8.534316096540877480763042876204, 10.02248943322612552983349154037, 10.81969285282630764680299922057