| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 4·7-s − 8-s + 9-s − 6·11-s − 2·12-s − 2·13-s + 4·14-s + 16-s − 18-s − 4·19-s + 8·21-s + 6·22-s + 2·24-s + 2·26-s + 4·27-s − 4·28-s + 4·31-s − 32-s + 12·33-s + 36-s − 4·37-s + 4·38-s + 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 1.74·21-s + 1.27·22-s + 0.408·24-s + 0.392·26-s + 0.769·27-s − 0.755·28-s + 0.718·31-s − 0.176·32-s + 2.08·33-s + 1/6·36-s − 0.657·37-s + 0.648·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.50514909362706, −15.99170167281328, −15.48664943359084, −15.10881179870668, −14.11823560512770, −13.23447963183461, −12.99611021317329, −12.36175420541600, −11.86528540951495, −11.21454397742475, −10.47985046240228, −10.14511080505373, −9.919103527317289, −8.827854200677093, −8.434721631247959, −7.546258705974526, −6.977654608850467, −6.391263149716337, −5.881655698070916, −5.213150120846833, −4.615165600426554, −3.397567117772909, −2.810908841118525, −2.039886381877294, −0.5623966478004139, 0,
0.5623966478004139, 2.039886381877294, 2.810908841118525, 3.397567117772909, 4.615165600426554, 5.213150120846833, 5.881655698070916, 6.391263149716337, 6.977654608850467, 7.546258705974526, 8.434721631247959, 8.827854200677093, 9.919103527317289, 10.14511080505373, 10.47985046240228, 11.21454397742475, 11.86528540951495, 12.36175420541600, 12.99611021317329, 13.23447963183461, 14.11823560512770, 15.10881179870668, 15.48664943359084, 15.99170167281328, 16.50514909362706
