L(s) = 1 | + i·2-s + (−1 + i)3-s − 4-s + (−1 − i)6-s − i·8-s − i·9-s + (1 + i)11-s + (1 − i)12-s + 16-s + i·17-s + 18-s − 2i·19-s + (−1 + i)22-s + (1 + i)24-s − i·25-s + ⋯ |
L(s) = 1 | + i·2-s + (−1 + i)3-s − 4-s + (−1 − i)6-s − i·8-s − i·9-s + (1 + i)11-s + (1 − i)12-s + 16-s + i·17-s + 18-s − 2i·19-s + (−1 + i)22-s + (1 + i)24-s − i·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4499568977\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4499568977\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 17 | \( 1 - iT \) |
good | 3 | \( 1 + (1 - i)T - iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-1 - i)T + iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 + 2iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (1 + i)T + iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−14.10829379018122111854593772233, −12.83260234817688541120987881577, −11.77449855381196807410731862227, −10.57382924601604411035752812235, −9.663511030670705522782666378170, −8.718488301401035354803204082389, −7.09243364214847237611653603262, −6.15309186342952232387332769817, −4.89202963436691088873700380710, −4.12555615364533978441904967912,
1.42085106340463447177544392325, 3.55644053736731572668032407830, 5.33625654831850659557344740679, 6.35973643123559432573863254683, 7.80649168776885473376423669888, 9.083110844705150311940738088582, 10.31820460079360129722653774897, 11.58875341989955402464879387225, 11.76561960022422668253953730470, 12.85715835520574200022236771097