L(s) = 1 | + 2-s − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s − 2·11-s − 3·12-s − 13-s + 14-s − 3·15-s + 16-s − 3·17-s + 6·18-s + 6·19-s + 20-s − 3·21-s − 2·22-s − 4·23-s − 3·24-s + 25-s − 26-s − 9·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s − 0.866·12-s − 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s + 1.37·19-s + 0.223·20-s − 0.654·21-s − 0.426·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 0.196·26-s − 1.73·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.376623136\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.376623136\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
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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
−13.88459942260273, −13.34889171587435, −12.69022580585021, −12.50103452241304, −11.77989884657988, −11.51150483255350, −11.16392839766929, −10.47962030486497, −10.06832656500573, −9.794216663315734, −8.924206998147368, −8.191303059006412, −7.559050748170960, −7.121118409213530, −6.491653644291432, −6.126927803204376, −5.549377663344742, −5.073456495632729, −4.829415225366951, −4.149442441530068, −3.476046625930246, −2.547793056268012, −2.023911621286095, −1.151566430122852, −0.5404128605738992,
0.5404128605738992, 1.151566430122852, 2.023911621286095, 2.547793056268012, 3.476046625930246, 4.149442441530068, 4.829415225366951, 5.073456495632729, 5.549377663344742, 6.126927803204376, 6.491653644291432, 7.121118409213530, 7.559050748170960, 8.191303059006412, 8.924206998147368, 9.794216663315734, 10.06832656500573, 10.47962030486497, 11.16392839766929, 11.51150483255350, 11.77989884657988, 12.50103452241304, 12.69022580585021, 13.34889171587435, 13.88459942260273