| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 6·11-s + 16-s + 4·17-s + 4·19-s + 20-s − 6·22-s + 25-s + 9·29-s + 4·31-s − 32-s − 4·34-s + 8·37-s − 4·38-s − 40-s − 5·41-s + 6·44-s + 3·47-s − 7·49-s − 50-s − 6·53-s + 6·55-s − 9·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.80·11-s + 1/4·16-s + 0.970·17-s + 0.917·19-s + 0.223·20-s − 1.27·22-s + 1/5·25-s + 1.67·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s + 1.31·37-s − 0.648·38-s − 0.158·40-s − 0.780·41-s + 0.904·44-s + 0.437·47-s − 49-s − 0.141·50-s − 0.824·53-s + 0.809·55-s − 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.559875129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.559875129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.26906902402669, −12.58725795256749, −12.14305963604903, −11.83867852163860, −11.31417013497929, −10.94471496524321, −10.07249981912732, −9.847103561262800, −9.584864216396816, −8.948182612837735, −8.473694095530571, −8.056806628503784, −7.438101796897233, −6.872909407066998, −6.466972192710133, −6.044782498118824, −5.507047258914314, −4.708713581073805, −4.356660775404165, −3.330043381543212, −3.270716535344162, −2.376768710491473, −1.670861066671756, −1.058804833622367, −0.7521285156432091,
0.7521285156432091, 1.058804833622367, 1.670861066671756, 2.376768710491473, 3.270716535344162, 3.330043381543212, 4.356660775404165, 4.708713581073805, 5.507047258914314, 6.044782498118824, 6.466972192710133, 6.872909407066998, 7.438101796897233, 8.056806628503784, 8.473694095530571, 8.948182612837735, 9.584864216396816, 9.847103561262800, 10.07249981912732, 10.94471496524321, 11.31417013497929, 11.83867852163860, 12.14305963604903, 12.58725795256749, 13.26906902402669
