L(s) = 1 | − 2-s + 1.14·3-s + 4-s + 5-s − 1.14·6-s + 7-s − 8-s − 1.68·9-s − 10-s + 1.14·12-s − 4.68·13-s − 14-s + 1.14·15-s + 16-s + 0.292·17-s + 1.68·18-s − 6.51·19-s + 20-s + 1.14·21-s + 2.85·23-s − 1.14·24-s + 25-s + 4.68·26-s − 5.37·27-s + 28-s + 1.43·29-s − 1.14·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.661·3-s + 0.5·4-s + 0.447·5-s − 0.468·6-s + 0.377·7-s − 0.353·8-s − 0.561·9-s − 0.316·10-s + 0.330·12-s − 1.29·13-s − 0.267·14-s + 0.295·15-s + 0.250·16-s + 0.0709·17-s + 0.397·18-s − 1.49·19-s + 0.223·20-s + 0.250·21-s + 0.595·23-s − 0.234·24-s + 0.200·25-s + 0.918·26-s − 1.03·27-s + 0.188·28-s + 0.267·29-s − 0.209·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.596359104\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.596359104\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 13 | \( 1 + 4.68T + 13T^{2} \) |
| 17 | \( 1 - 0.292T + 17T^{2} \) |
| 19 | \( 1 + 6.51T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 - 1.43T + 29T^{2} \) |
| 31 | \( 1 - 0.978T + 31T^{2} \) |
| 37 | \( 1 - 0.853T + 37T^{2} \) |
| 41 | \( 1 - 6.22T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 9.95T + 47T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 + 9.37T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 0.585T + 67T^{2} \) |
| 71 | \( 1 + 0.335T + 71T^{2} \) |
| 73 | \( 1 - 3.70T + 73T^{2} \) |
| 79 | \( 1 - 2.51T + 79T^{2} \) |
| 83 | \( 1 - 1.70T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 9.10T + 97T^{2} \) |
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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
−7.903712515451102369404979111736, −7.31419616245497645615733726648, −6.52140951385644753733343816993, −5.84907127756862277030340796421, −5.01786326332934573911360804580, −4.26606229740453695291208601117, −3.17613585520047066847966027986, −2.40910225422251583507326352964, −1.99427337453769251688043633450, −0.63596818524233915662653421223,
0.63596818524233915662653421223, 1.99427337453769251688043633450, 2.40910225422251583507326352964, 3.17613585520047066847966027986, 4.26606229740453695291208601117, 5.01786326332934573911360804580, 5.84907127756862277030340796421, 6.52140951385644753733343816993, 7.31419616245497645615733726648, 7.903712515451102369404979111736