L(s) = 1 | + 3.24i·3-s − 0.276i·5-s + 5.07i·7-s − 7.50·9-s + 3.78i·11-s + 3.39·13-s + 0.896·15-s + (0.332 − 4.10i)17-s − 7.59·19-s − 16.4·21-s + 8.61i·23-s + 4.92·25-s − 14.6i·27-s − 8.08i·29-s − 6.90i·31-s + ⋯ |
L(s) = 1 | + 1.87i·3-s − 0.123i·5-s + 1.91i·7-s − 2.50·9-s + 1.14i·11-s + 0.942·13-s + 0.231·15-s + (0.0805 − 0.996i)17-s − 1.74·19-s − 3.58·21-s + 1.79i·23-s + 0.984·25-s − 2.81i·27-s − 1.50i·29-s − 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0805 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0805 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.074922411\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.074922411\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-0.332 + 4.10i)T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 3.24iT - 3T^{2} \) |
| 5 | \( 1 + 0.276iT - 5T^{2} \) |
| 7 | \( 1 - 5.07iT - 7T^{2} \) |
| 11 | \( 1 - 3.78iT - 11T^{2} \) |
| 13 | \( 1 - 3.39T + 13T^{2} \) |
| 19 | \( 1 + 7.59T + 19T^{2} \) |
| 23 | \( 1 - 8.61iT - 23T^{2} \) |
| 29 | \( 1 + 8.08iT - 29T^{2} \) |
| 31 | \( 1 + 6.90iT - 31T^{2} \) |
| 37 | \( 1 - 5.47iT - 37T^{2} \) |
| 41 | \( 1 - 5.51iT - 41T^{2} \) |
| 43 | \( 1 + 4.87T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 + 3.84T + 53T^{2} \) |
| 61 | \( 1 - 12.2iT - 61T^{2} \) |
| 67 | \( 1 - 8.67T + 67T^{2} \) |
| 71 | \( 1 - 0.288iT - 71T^{2} \) |
| 73 | \( 1 - 1.40iT - 73T^{2} \) |
| 79 | \( 1 + 11.0iT - 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 - 3.59T + 89T^{2} \) |
| 97 | \( 1 - 0.104iT - 97T^{2} \) |
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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
−9.072964039003629759922970028912, −8.577146915335156140902600507273, −7.87867792131632054929308056071, −6.37290056188270643141437170029, −5.89792364751574637437851302909, −5.02477649633778901431484487199, −4.64737363458540344827154389495, −3.70439912636255432756455250709, −2.82586518356345647621004120638, −2.06195110421515693327517631834,
0.33013959213271772317237402432, 1.09265779242630714495834290690, 1.92141711508200708663675921444, 3.19229454933149424319408736627, 3.81077474901099595955970663063, 4.98309428582679629863438758647, 6.18312792999556536894909327058, 6.71106051503118243440310600666, 6.83245078946661826480529654044, 7.995436463534311589396359741628