| L(s) = 1 | + (1.04 − 1.37i)3-s − 5-s + 2.10i·7-s + (−0.796 − 2.89i)9-s + 2.45·11-s − 5.70·13-s + (−1.04 + 1.37i)15-s + 6.65·17-s − 4.23i·19-s + (2.89 + 2.20i)21-s + (3.63 − 3.12i)23-s + 25-s + (−4.82 − 1.93i)27-s − 10.0i·29-s + 4.21·31-s + ⋯ |
| L(s) = 1 | + (0.606 − 0.795i)3-s − 0.447·5-s + 0.793i·7-s + (−0.265 − 0.964i)9-s + 0.741·11-s − 1.58·13-s + (−0.271 + 0.355i)15-s + 1.61·17-s − 0.970i·19-s + (0.631 + 0.481i)21-s + (0.758 − 0.651i)23-s + 0.200·25-s + (−0.927 − 0.373i)27-s − 1.86i·29-s + 0.757·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0580 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0580 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.752728397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.752728397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.04 + 1.37i)T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-3.63 + 3.12i)T \) |
| good | 7 | \( 1 - 2.10iT - 7T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 - 6.65T + 17T^{2} \) |
| 19 | \( 1 + 4.23iT - 19T^{2} \) |
| 29 | \( 1 + 10.0iT - 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 37 | \( 1 + 8.14iT - 37T^{2} \) |
| 41 | \( 1 + 3.58iT - 41T^{2} \) |
| 43 | \( 1 + 2.94iT - 43T^{2} \) |
| 47 | \( 1 - 6.21iT - 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 6.31iT - 59T^{2} \) |
| 61 | \( 1 - 6.35iT - 61T^{2} \) |
| 67 | \( 1 - 11.5iT - 67T^{2} \) |
| 71 | \( 1 + 4.67iT - 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 8.63iT - 79T^{2} \) |
| 83 | \( 1 - 0.357T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 3.20iT - 97T^{2} \) |
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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
−9.361488014533743460653374463991, −8.507546004026289971769986319640, −7.72014132265967179329939473542, −7.13379041869747119717023911698, −6.22825908997228873259768580118, −5.25515905156195493501610514619, −4.15095615974093641808551668896, −2.95298566825214488168620620324, −2.27946416184367774150667958286, −0.71472224125583373196289079461,
1.40359690476898235276620023458, 3.06636727118958190970479060976, 3.60308827150465261069009065208, 4.67739299656867472109781266118, 5.26883154330284938561020761977, 6.69237659464284792898771495262, 7.59099793166016637729504756435, 8.026736849175491619851990316793, 9.117486417707077504566303592348, 9.864108687506164806084866769889
