L(s) = 1 | + 2-s + 4-s + (−2 − i)5-s − 3i·7-s + 8-s + (−2 − i)10-s + 3·11-s − 4·13-s − 3i·14-s + 16-s − 7·17-s + 4i·19-s + (−2 − i)20-s + 3·22-s − 6·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.894 − 0.447i)5-s − 1.13i·7-s + 0.353·8-s + (−0.632 − 0.316i)10-s + 0.904·11-s − 1.10·13-s − 0.801i·14-s + 0.250·16-s − 1.69·17-s + 0.917i·19-s + (−0.447 − 0.223i)20-s + 0.639·22-s − 1.25·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 37 | \( 1 + (-6 - i)T \) |
good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 5iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−7.956555329998575745283673494655, −7.41021361158273499063481624693, −6.69920253220864371515547146483, −5.96997046907787308616731441224, −4.74289753763430731219637328693, −4.21866124720168452169509188987, −3.85232890071935981437975915187, −2.58452729523013995160084059365, −1.38628598957560626554783562742, 0,
2.03598916193688812775897295309, 2.67003249263391527849169126892, 3.67240067800049302071045514792, 4.47838947328881990374318444354, 5.09850215650327575403620761412, 6.16840648165814891284467185500, 6.76706723812939217514339957131, 7.35996393949692343753206835153, 8.359868199944863199021240091301