L(s) = 1 | − 2-s + 4-s + i·5-s − 8-s + i·9-s − i·10-s − i·13-s + 16-s + (1 + i)17-s − i·18-s + i·20-s − 25-s + i·26-s − 2i·29-s − 32-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + i·5-s − 8-s + i·9-s − i·10-s − i·13-s + 16-s + (1 + i)17-s − i·18-s + i·20-s − 25-s + i·26-s − 2i·29-s − 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5059126929\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5059126929\) |
\(L(1)\) |
|
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 | \( 1 - iT \) |
| 13 | \( 1 + iT \) |
good | 3 | \( 1 - iT^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + 2iT - T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1 + i)T + iT^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 - 2T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-1 - i)T + iT^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−12.02582929086622292141981981243, −11.04239992511272870701939745140, −10.37317329210454955630126998659, −9.749092543078072203135029193729, −8.109643467058337085922005046852, −7.81045689213456306064817317062, −6.53443278402019860554475465853, −5.53657516302876886290621075646, −3.44649809219711510079715086898, −2.16114574364275784524204190894,
1.37736054286372667276825657430, 3.33858863379832676236069890591, 5.00857670055161515029135103719, 6.32641308821268774433648266607, 7.32346580397877620284854966713, 8.478296824641379647754730030440, 9.274106424248876804879528439435, 9.815536100597621499556067821481, 11.20980221048105044725047029261, 12.03868078674636152471222880535