L(s) = 1 | + i·3-s − 2.23·5-s + i·7-s − 9-s + 2·11-s + 4.47i·13-s − 2.23i·15-s + 2.47i·17-s + 2·19-s − 21-s − 4i·23-s + 5.00·25-s − i·27-s − 0.472·29-s − 8.47·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.999·5-s + 0.377i·7-s − 0.333·9-s + 0.603·11-s + 1.24i·13-s − 0.577i·15-s + 0.599i·17-s + 0.458·19-s − 0.218·21-s − 0.834i·23-s + 1.00·25-s − 0.192i·27-s − 0.0876·29-s − 1.52·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5934592540\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5934592540\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 4.47iT - 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 - 6.47iT - 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 6.47iT - 43T^{2} \) |
| 47 | \( 1 + 2.47iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 10.4iT - 67T^{2} \) |
| 71 | \( 1 + 3.52T + 71T^{2} \) |
| 73 | \( 1 - 16.4iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 9.41T + 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
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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
−9.693278842040134611006482744339, −8.746497360403312809373449668803, −8.519708799917779105595930343976, −7.27392057825576723112719927133, −6.67427514917933090639374342942, −5.60666589712115886154968881518, −4.59429385487275223150575003965, −3.96408555542793988421582208168, −3.11286114229709259824528097070, −1.67740086912831011580957148508,
0.23296643099176709284440640549, 1.48169200962818908037988918695, 3.07479837211408391179632462678, 3.66964663581635087881691048687, 4.84800183149000539712994214537, 5.69562896916051872321719933051, 6.77231091253825688547638943598, 7.50528284398475001564572785446, 7.895785411946870728288897305538, 8.874709829274241576651480175042