| L(s) = 1 | + 3-s − i·5-s + i·7-s − 2·9-s − 3i·11-s − i·15-s + 3·17-s + 5i·19-s + i·21-s − 9·23-s − 25-s − 5·27-s − 9·29-s + 8i·31-s − 3i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447i·5-s + 0.377i·7-s − 0.666·9-s − 0.904i·11-s − 0.258i·15-s + 0.727·17-s + 1.14i·19-s + 0.218i·21-s − 1.87·23-s − 0.200·25-s − 0.962·27-s − 1.67·29-s + 1.43i·31-s − 0.522i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7989904894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7989904894\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 5iT - 19T^{2} \) |
| 23 | \( 1 + 9T + 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 - 3iT - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 9iT - 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 - 9iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 3iT - 89T^{2} \) |
| 97 | \( 1 - 17iT - 97T^{2} \) |
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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
−8.747822336928920243967286541707, −8.119835155430687636170451690378, −7.79671815510750244378478594292, −6.50650617511480785901082878294, −5.68715628429157400917095491878, −5.35379742121607885638979907732, −3.94036531501304038414612871862, −3.43727263550882673983140698158, −2.41956503605614468970415125290, −1.39743196763718808093819162818,
0.20794397042005058094806288444, 1.97565966585360190061995054872, 2.57347257545187325681429332499, 3.71554123870115520137958058362, 4.20605763088023793278058537453, 5.48150112663575078687124018658, 6.00701714687442570923136873491, 7.15942214953401994021429701168, 7.55935750465578414101877219972, 8.240828408226520802480695744429
