L(s) = 1 | + i·2-s − 4-s + 2i·5-s − i·7-s − i·8-s − 2·10-s + (2 − 3i)13-s + 14-s + 16-s + 2·17-s − 4i·19-s − 2i·20-s + 6·23-s + 25-s + (3 + 2i)26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.894i·5-s − 0.377i·7-s − 0.353i·8-s − 0.632·10-s + (0.554 − 0.832i)13-s + 0.267·14-s + 0.250·16-s + 0.485·17-s − 0.917i·19-s − 0.447i·20-s + 1.25·23-s + 0.200·25-s + (0.588 + 0.392i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.690956669\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.690956669\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-2 + 3i)T \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.424698992451957286159519272060, −8.595071690056888259044084140301, −7.75638654433286324095934684822, −7.06286264566425336447786157760, −6.45353718205156970366882323021, −5.52967677766559342397443899790, −4.66830019815387131361063796642, −3.50908506190115822744657955592, −2.76494907188270627237837438418, −0.924820628258119128540482393297,
0.977408154245149686882164570202, 1.97509705869079714685717712568, 3.24666959081167690420024510323, 4.16199777834455602780680036225, 5.04112035904256504823063621828, 5.77219638747517919045293616973, 6.84202525551738398982228498201, 7.934325375544616487882788847408, 8.749168164747724714042925181899, 9.152418986267061821224835424456