| L(s) = 1 | − 1.00i·3-s + 1.31i·5-s + (1.40 + 2.24i)7-s + 1.98·9-s + (−2.38 − 2.30i)11-s − 13-s + 1.32·15-s + 1.70·17-s − 7.19·19-s + (2.25 − 1.41i)21-s − 3.72·23-s + 3.27·25-s − 5.01i·27-s − 7.88i·29-s − 5.82i·31-s + ⋯ |
| L(s) = 1 | − 0.581i·3-s + 0.587i·5-s + (0.529 + 0.848i)7-s + 0.662·9-s + (−0.717 − 0.696i)11-s − 0.277·13-s + 0.341·15-s + 0.412·17-s − 1.65·19-s + (0.492 − 0.307i)21-s − 0.777·23-s + 0.655·25-s − 0.965i·27-s − 1.46i·29-s − 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.292560807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292560807\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.40 - 2.24i)T \) |
| 11 | \( 1 + (2.38 + 2.30i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.00iT - 3T^{2} \) |
| 5 | \( 1 - 1.31iT - 5T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 23 | \( 1 + 3.72T + 23T^{2} \) |
| 29 | \( 1 + 7.88iT - 29T^{2} \) |
| 31 | \( 1 + 5.82iT - 31T^{2} \) |
| 37 | \( 1 + 8.84T + 37T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 43 | \( 1 + 7.07iT - 43T^{2} \) |
| 47 | \( 1 + 3.62iT - 47T^{2} \) |
| 53 | \( 1 - 6.34T + 53T^{2} \) |
| 59 | \( 1 - 1.65iT - 59T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 7.55T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 12.8iT - 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 10.7iT - 89T^{2} \) |
| 97 | \( 1 - 1.83iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.355078696547414196985433139592, −7.50838780476309001573689152952, −6.78872701237528529234713053451, −6.07962327029449243020106215060, −5.40807031166734605309780736942, −4.46026529468280052428024014139, −3.54097191046463806826661632068, −2.33412150375930489071848583981, −2.00826169637204504641430436231, −0.37031259940925177388451543393,
1.22183294495730976164042726485, 2.10568020155431073122298496299, 3.47661923677165776024101993487, 4.20695784823669711128114565444, 4.94876819472423017485754380561, 5.22504789474989037182828226280, 6.77236953230098754287711503381, 7.03587388908068240221717522924, 8.126509075268390758290321361513, 8.496877272302489268699911072287
