| L(s) = 1 | + 2.30i·3-s − 0.586i·5-s + (−2.63 + 0.279i)7-s − 2.32·9-s + (0.922 + 3.18i)11-s + 13-s + 1.35·15-s + 3.52·17-s + 8.50·19-s + (−0.645 − 6.07i)21-s − 4.27·23-s + 4.65·25-s + 1.55i·27-s + 7.21i·29-s − 8.28i·31-s + ⋯ |
| L(s) = 1 | + 1.33i·3-s − 0.262i·5-s + (−0.994 + 0.105i)7-s − 0.775·9-s + (0.277 + 0.960i)11-s + 0.277·13-s + 0.349·15-s + 0.855·17-s + 1.95·19-s + (−0.140 − 1.32i)21-s − 0.892·23-s + 0.931·25-s + 0.298i·27-s + 1.34i·29-s − 1.48i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.850966555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.850966555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.279i)T \) |
| 11 | \( 1 + (-0.922 - 3.18i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.30iT - 3T^{2} \) |
| 5 | \( 1 + 0.586iT - 5T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 - 8.50T + 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 - 7.21iT - 29T^{2} \) |
| 31 | \( 1 + 8.28iT - 31T^{2} \) |
| 37 | \( 1 - 4.64T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 8.25iT - 43T^{2} \) |
| 47 | \( 1 + 0.348iT - 47T^{2} \) |
| 53 | \( 1 + 8.42T + 53T^{2} \) |
| 59 | \( 1 + 10.8iT - 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 4.54T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 16.9iT - 89T^{2} \) |
| 97 | \( 1 - 11.1iT - 97T^{2} \) |
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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
−9.164013848130522749598831644967, −7.930722216569534454408462585505, −7.27736965705844930815750477927, −6.36925323246754336740813530507, −5.46267249774730569828829854108, −4.98096325466772780515741473312, −3.94491409504936120821368637782, −3.53505819544191959460315130573, −2.56220589666754339232990706390, −1.05016694640500828640256207176,
0.68654254361372493777164039901, 1.37912758392095846544827711354, 2.86367370622000301280617894143, 3.16339211432296813659703462353, 4.31902916541665188054025017581, 5.78114988537909228317464593180, 5.96592916455724463877245518858, 6.82165257501100446919107714554, 7.44930352640258856405657827224, 7.986442118780826834902676379270
