L(s) = 1 | + 3-s + i·4-s + 9-s − i·11-s + i·12-s − 16-s + (−1 + i)23-s + 27-s − i·33-s + i·36-s + (−1 + i)37-s + 44-s + (−1 − i)47-s − 48-s − i·49-s + ⋯ |
L(s) = 1 | + 3-s + i·4-s + 9-s − i·11-s + i·12-s − 16-s + (−1 + i)23-s + 27-s − i·33-s + i·36-s + (−1 + i)37-s + 44-s + (−1 − i)47-s − 48-s − i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.333529343\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.333529343\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + iT \) |
good | 2 | \( 1 - iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (1 + i)T + iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 + 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-1 + i)T - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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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
−10.36571778686003580948140923142, −9.481257921491071449289343722474, −8.591804656229814635110589604261, −8.142041314699648138692443863417, −7.31486190148978526931149268542, −6.38356191354787425437317710929, −4.98487532427835413634720141374, −3.72826027780345103902579261309, −3.26117312805604358064281287204, −1.98688581388910472195832161351,
1.63065886993620089063110655326, 2.59270400152186738378666369983, 4.08101112241101502930218202097, 4.83849572637896556163742031404, 6.06922225315252916403817368698, 6.97667592045900352333327819068, 7.81405187021998106449064123698, 8.841592491818005228080455450476, 9.525740176187799810310843416756, 10.21163218496915634897103937202