L(s) = 1 | − i·3-s + (2.23 − 0.153i)5-s − 3.71i·7-s − 9-s + 3.31·11-s − i·13-s + (−0.153 − 2.23i)15-s + 1.55i·17-s + 5.33·19-s − 3.71·21-s − 0.442i·23-s + (4.95 − 0.683i)25-s + i·27-s − 2.56·29-s + 0.613·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.997 − 0.0685i)5-s − 1.40i·7-s − 0.333·9-s + 1.00·11-s − 0.277i·13-s + (−0.0395 − 0.575i)15-s + 0.377i·17-s + 1.22·19-s − 0.810·21-s − 0.0923i·23-s + (0.990 − 0.136i)25-s + 0.192i·27-s − 0.477·29-s + 0.110·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0685 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0685 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.157560283\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.157560283\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.23 + 0.153i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 + 3.71iT - 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 17 | \( 1 - 1.55iT - 17T^{2} \) |
| 19 | \( 1 - 5.33T + 19T^{2} \) |
| 23 | \( 1 + 0.442iT - 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 - 0.613T + 31T^{2} \) |
| 37 | \( 1 + 0.257iT - 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 12.6iT - 43T^{2} \) |
| 47 | \( 1 - 7.44iT - 47T^{2} \) |
| 53 | \( 1 - 5.39iT - 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 5.27T + 61T^{2} \) |
| 67 | \( 1 - 10.5iT - 67T^{2} \) |
| 71 | \( 1 - 0.311T + 71T^{2} \) |
| 73 | \( 1 + 9.46iT - 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 11.7iT - 83T^{2} \) |
| 89 | \( 1 - 4.58T + 89T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−9.276372292068161598679187105586, −8.485046205746301367181333624581, −7.38776594924415378039442250554, −6.94737362347609535634054883861, −6.08141495558636714983690707375, −5.24468359162913408218840503222, −4.10770072186520551300212112739, −3.17244077604370045869243839262, −1.76012893497043518970983295314, −0.924579956646139943493053418109,
1.55076479165597003136262703128, 2.62033463441366038360608723668, 3.53701145262437217623485004538, 4.88250898969726350993430997782, 5.47845275068968321897050019226, 6.23208458518780412615188862935, 7.02967938598617260548417113390, 8.372093994606482963468766149159, 9.005722236259769652171816076944, 9.624008374226867119130863789933