| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 4.10·7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 1.10·13-s − 4.10·14-s − 15-s + 16-s − 17-s + 18-s + 1.10·19-s + 20-s + 4.10·21-s + 22-s + 2.57·23-s − 24-s + 25-s + 1.10·26-s − 27-s − 4.10·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.55·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.307·13-s − 1.09·14-s − 0.258·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.254·19-s + 0.223·20-s + 0.896·21-s + 0.213·22-s + 0.537·23-s − 0.204·24-s + 0.200·25-s + 0.217·26-s − 0.192·27-s − 0.776·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287214708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287214708\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4.10T + 7T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 19 | \( 1 - 1.10T + 19T^{2} \) |
| 23 | \( 1 - 2.57T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 - 0.184T + 31T^{2} \) |
| 37 | \( 1 + 1.86T + 37T^{2} \) |
| 41 | \( 1 + 4.68T + 41T^{2} \) |
| 43 | \( 1 + 9.72T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 1.45T + 59T^{2} \) |
| 61 | \( 1 - 6.81T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 4.76T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 + 6.33T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 2.40T + 97T^{2} \) |
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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
−8.089862602689667851637515479164, −6.86040949984687844673979768772, −6.64337416308061099783143883307, −6.12542236125703336834272131588, −5.23503738881987562652044506176, −4.64197829843350269968436554912, −3.52407031428271906479744586055, −3.11298923736282601472141992445, −1.97712668131625903310120744495, −0.73889730628603333872538244247,
0.73889730628603333872538244247, 1.97712668131625903310120744495, 3.11298923736282601472141992445, 3.52407031428271906479744586055, 4.64197829843350269968436554912, 5.23503738881987562652044506176, 6.12542236125703336834272131588, 6.64337416308061099783143883307, 6.86040949984687844673979768772, 8.089862602689667851637515479164
