L(s) = 1 | + i·3-s + (−2.17 + 0.539i)5-s − i·7-s − 9-s − 5.41·11-s − 4.34i·13-s + (−0.539 − 2.17i)15-s + 1.07i·17-s + 4.34·19-s + 21-s + 6.34i·23-s + (4.41 − 2.34i)25-s − i·27-s + 8.83·29-s + 4.34·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.970 + 0.241i)5-s − 0.377i·7-s − 0.333·9-s − 1.63·11-s − 1.20i·13-s + (−0.139 − 0.560i)15-s + 0.261i·17-s + 0.995·19-s + 0.218·21-s + 1.32i·23-s + (0.883 − 0.468i)25-s − 0.192i·27-s + 1.64·29-s + 0.779·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.145007851\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145007851\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.17 - 0.539i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 + 4.34iT - 13T^{2} \) |
| 17 | \( 1 - 1.07iT - 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 - 6.34iT - 23T^{2} \) |
| 29 | \( 1 - 8.83T + 29T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 + 8.68iT - 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 - 6.15iT - 43T^{2} \) |
| 47 | \( 1 - 6.83iT - 47T^{2} \) |
| 53 | \( 1 + 6.18iT - 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 11.1iT - 73T^{2} \) |
| 79 | \( 1 - 0.680T + 79T^{2} \) |
| 83 | \( 1 + 6.83iT - 83T^{2} \) |
| 89 | \( 1 + 6.49T + 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−9.564277994883946461572060717930, −8.336539527887057452676370156262, −7.83177369883951929111866724786, −7.32239942961192359820609003466, −5.98570045027623519413799006748, −5.18120823764302407220994672912, −4.41724729922782402160497539854, −3.30532378375323121051098202255, −2.78684825010035653851783874512, −0.68641881922207452473164270519,
0.76941064174867645652252405683, 2.36692912680192085570046013664, 3.13613461209930975958805121364, 4.53436903673034710251402601475, 5.03956177798313009773268344023, 6.24079085039635272692256266534, 7.02144422522147266905429286974, 7.83102584252278855298722402817, 8.357947175600839228603679654785, 9.105383702227852249205920103744