| L(s) = 1 | + 3.15i·3-s + 2.09i·5-s + (1.59 + 2.10i)7-s − 6.92·9-s + (1.48 + 2.96i)11-s + 13-s − 6.61·15-s + 5.21·17-s − 3.50·19-s + (−6.64 + 5.03i)21-s − 3.97·23-s + 0.593·25-s − 12.3i·27-s + 0.171i·29-s − 6.22i·31-s + ⋯ |
| L(s) = 1 | + 1.81i·3-s + 0.938i·5-s + (0.604 + 0.796i)7-s − 2.30·9-s + (0.447 + 0.894i)11-s + 0.277·13-s − 1.70·15-s + 1.26·17-s − 0.804·19-s + (−1.44 + 1.09i)21-s − 0.828·23-s + 0.118·25-s − 2.37i·27-s + 0.0317i·29-s − 1.11i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.511115876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511115876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.59 - 2.10i)T \) |
| 11 | \( 1 + (-1.48 - 2.96i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 3.15iT - 3T^{2} \) |
| 5 | \( 1 - 2.09iT - 5T^{2} \) |
| 17 | \( 1 - 5.21T + 17T^{2} \) |
| 19 | \( 1 + 3.50T + 19T^{2} \) |
| 23 | \( 1 + 3.97T + 23T^{2} \) |
| 29 | \( 1 - 0.171iT - 29T^{2} \) |
| 31 | \( 1 + 6.22iT - 31T^{2} \) |
| 37 | \( 1 + 9.42T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 - 2.92iT - 43T^{2} \) |
| 47 | \( 1 - 9.40iT - 47T^{2} \) |
| 53 | \( 1 - 5.09T + 53T^{2} \) |
| 59 | \( 1 + 9.32iT - 59T^{2} \) |
| 61 | \( 1 + 0.809T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 2.13iT - 79T^{2} \) |
| 83 | \( 1 + 4.96T + 83T^{2} \) |
| 89 | \( 1 + 15.0iT - 89T^{2} \) |
| 97 | \( 1 - 1.01iT - 97T^{2} \) |
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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
−9.089084674946956511078668873042, −8.382055019165394018627883947769, −7.65913911046667560192773451477, −6.53745543899193477406323645634, −5.82101409607505573955884368308, −5.11157292254228387137541031135, −4.37261591433937157716838915759, −3.65309459744773740548801321376, −2.89563267781488502767694618649, −1.92792792592164537440543353021,
0.44253270431124290950172564485, 1.26631574295664556537313772084, 1.81280454663098302005882279893, 3.16100204627896314999561938953, 4.04001205229161573698529465805, 5.27860911088327231234453660184, 5.71058661609835352586193165041, 6.76005704016722828794453467753, 7.10105386034108114348089321239, 8.164490409427792937858005023208
