| L(s) = 1 | + 0.968i·3-s + 0.815i·5-s + (−2.53 − 0.768i)7-s + 2.06·9-s + (0.542 − 3.27i)11-s + 13-s − 0.789·15-s − 3.02·17-s − 2.23·19-s + (0.744 − 2.45i)21-s − 0.0738·23-s + 4.33·25-s + 4.90i·27-s + 2.16i·29-s + 0.394i·31-s + ⋯ |
| L(s) = 1 | + 0.558i·3-s + 0.364i·5-s + (−0.956 − 0.290i)7-s + 0.687·9-s + (0.163 − 0.986i)11-s + 0.277·13-s − 0.203·15-s − 0.733·17-s − 0.512·19-s + (0.162 − 0.534i)21-s − 0.0153·23-s + 0.867·25-s + 0.943i·27-s + 0.401i·29-s + 0.0707i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5422127603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5422127603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.53 + 0.768i)T \) |
| 11 | \( 1 + (-0.542 + 3.27i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.968iT - 3T^{2} \) |
| 5 | \( 1 - 0.815iT - 5T^{2} \) |
| 17 | \( 1 + 3.02T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 0.0738T + 23T^{2} \) |
| 29 | \( 1 - 2.16iT - 29T^{2} \) |
| 31 | \( 1 - 0.394iT - 31T^{2} \) |
| 37 | \( 1 + 5.14T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 5.58iT - 43T^{2} \) |
| 47 | \( 1 + 6.69iT - 47T^{2} \) |
| 53 | \( 1 - 0.252T + 53T^{2} \) |
| 59 | \( 1 + 5.71iT - 59T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 9.86T + 71T^{2} \) |
| 73 | \( 1 + 3.58T + 73T^{2} \) |
| 79 | \( 1 - 9.98iT - 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 2.33iT - 89T^{2} \) |
| 97 | \( 1 + 8.67iT - 97T^{2} \) |
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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
−8.541645229296365882489605887325, −7.17601935367406540507503483777, −6.83625862746288925102758862308, −6.09303315734587215144889742708, −5.18288035660930494912369196921, −4.26439678962198251374940323756, −3.55074214734986768845669054920, −2.95143070155342935074759838832, −1.59795222974677912942711370876, −0.15562961414543632233229206373,
1.30296282994315079034697149388, 2.17313210825110903205824496330, 3.17329866925345859325476420845, 4.25974320970996246526003620234, 4.78442777802240892896578514344, 5.94327140535394196777884658254, 6.60947019789025828347377481074, 7.07458153326940114591590722263, 7.86308728140574257909146375086, 8.827450622765461409008728758537
