| L(s) = 1 | + (−2.14 + 2.09i)3-s − 4.10i·5-s − 9.08·7-s + (0.233 − 8.99i)9-s + 17.0·13-s + (8.59 + 8.81i)15-s − 10.7i·17-s + 9.28·19-s + (19.5 − 19.0i)21-s + 39.8i·23-s + 8.15·25-s + (18.3 + 19.8i)27-s + 16.5i·29-s + 12.0·31-s + 37.2i·35-s + ⋯ |
| L(s) = 1 | + (−0.716 + 0.697i)3-s − 0.820i·5-s − 1.29·7-s + (0.0259 − 0.999i)9-s + 1.31·13-s + (0.572 + 0.587i)15-s − 0.630i·17-s + 0.488·19-s + (0.929 − 0.905i)21-s + 1.73i·23-s + 0.326·25-s + (0.679 + 0.734i)27-s + 0.571i·29-s + 0.390·31-s + 1.06i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4385765569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4385765569\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.14 - 2.09i)T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 4.10iT - 25T^{2} \) |
| 7 | \( 1 + 9.08T + 49T^{2} \) |
| 13 | \( 1 - 17.0T + 169T^{2} \) |
| 17 | \( 1 + 10.7iT - 289T^{2} \) |
| 19 | \( 1 - 9.28T + 361T^{2} \) |
| 23 | \( 1 - 39.8iT - 529T^{2} \) |
| 29 | \( 1 - 16.5iT - 841T^{2} \) |
| 31 | \( 1 - 12.0T + 961T^{2} \) |
| 37 | \( 1 + 26.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 50.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 77.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 74.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 40.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 9.45T + 3.72e3T^{2} \) |
| 67 | \( 1 - 37.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 79.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 39.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 92.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 58.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 103.T + 9.40e3T^{2} \) |
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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
−9.073145704958678583070886724189, −8.609924471744035442689705829076, −7.18511811070823171552113592448, −6.47198350274182656485073379304, −5.55699531209435575413632083189, −5.03271093173498977213383700422, −3.73218099712560372306774523745, −3.30762785537968549184480871595, −1.31423734774577305159796325628, −0.15817545215852309582524599452,
1.18999151652844131382206657905, 2.62788799147715829996720522240, 3.43844560078172966516057067083, 4.64463813997163896617509475824, 5.97695978739094299562276638877, 6.39363630709374154544698696005, 6.86082619175016310579180733977, 7.949994133080703467453549471047, 8.715020003189528916069210576169, 9.863611678002573917830136262840
