L(s) = 1 | + (−0.826 − 0.563i)4-s + (0.222 + 0.974i)7-s + (−0.290 + 0.0663i)13-s + (0.365 + 0.930i)16-s + (1.68 + 0.974i)19-s + (0.955 − 0.294i)25-s + (0.365 − 0.930i)28-s + (0.975 − 0.563i)31-s + (−1.36 + 0.930i)37-s + (0.914 − 1.14i)43-s + (−0.900 + 0.433i)49-s + (0.277 + 0.108i)52-s + (1.12 + 1.64i)61-s + (0.222 − 0.974i)64-s + (−0.365 − 0.632i)67-s + ⋯ |
L(s) = 1 | + (−0.826 − 0.563i)4-s + (0.222 + 0.974i)7-s + (−0.290 + 0.0663i)13-s + (0.365 + 0.930i)16-s + (1.68 + 0.974i)19-s + (0.955 − 0.294i)25-s + (0.365 − 0.930i)28-s + (0.975 − 0.563i)31-s + (−1.36 + 0.930i)37-s + (0.914 − 1.14i)43-s + (−0.900 + 0.433i)49-s + (0.277 + 0.108i)52-s + (1.12 + 1.64i)61-s + (0.222 − 0.974i)64-s + (−0.365 − 0.632i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9237606852\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9237606852\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.222 - 0.974i)T \) |
good | 2 | \( 1 + (0.826 + 0.563i)T^{2} \) |
| 5 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 11 | \( 1 + (0.0747 - 0.997i)T^{2} \) |
| 13 | \( 1 + (0.290 - 0.0663i)T + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 19 | \( 1 + (-1.68 - 0.974i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.988 - 0.149i)T^{2} \) |
| 29 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.975 + 0.563i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1.36 - 0.930i)T + (0.365 - 0.930i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (-0.914 + 1.14i)T + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 53 | \( 1 + (0.365 + 0.930i)T^{2} \) |
| 59 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 61 | \( 1 + (-1.12 - 1.64i)T + (-0.365 + 0.930i)T^{2} \) |
| 67 | \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.255 - 0.829i)T + (-0.826 + 0.563i)T^{2} \) |
| 79 | \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 97 | \( 1 + 1.36iT - T^{2} \) |
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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
−9.881747358662268316654978905816, −9.023679238061335597644171027256, −8.471362025940182723911915456803, −7.55920241052307615321388795707, −6.41060011562060407138241285792, −5.48442292384963688751074859250, −5.04545239566626612603419144522, −3.90780681207675953569673776379, −2.72279859696078191042970372981, −1.33064800379332359694997552608,
0.985723670048946161206812940610, 2.88361662340205635903268130157, 3.71636542627490656687121619357, 4.74615908787407314768066254422, 5.25872408813903324047767747365, 6.76523993304005520299763098574, 7.41419678677991474660651531961, 8.105586055836899190781623387506, 9.051372175287564405865684301322, 9.647045546019426039142752429330