L(s) = 1 | + 2-s + 0.444·3-s + 4-s − 5-s + 0.444·6-s − 0.0224·7-s + 8-s − 2.80·9-s − 10-s − 1.62·11-s + 0.444·12-s − 13-s − 0.0224·14-s − 0.444·15-s + 16-s + 3.27·17-s − 2.80·18-s − 1.74·19-s − 20-s − 0.0100·21-s − 1.62·22-s + 1.12·23-s + 0.444·24-s + 25-s − 26-s − 2.58·27-s − 0.0224·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.256·3-s + 0.5·4-s − 0.447·5-s + 0.181·6-s − 0.00849·7-s + 0.353·8-s − 0.934·9-s − 0.316·10-s − 0.488·11-s + 0.128·12-s − 0.277·13-s − 0.00600·14-s − 0.114·15-s + 0.250·16-s + 0.793·17-s − 0.660·18-s − 0.400·19-s − 0.223·20-s − 0.00218·21-s − 0.345·22-s + 0.235·23-s + 0.0907·24-s + 0.200·25-s − 0.196·26-s − 0.496·27-s − 0.00424·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 - 0.444T + 3T^{2} \) |
| 7 | \( 1 + 0.0224T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 + 0.442T + 29T^{2} \) |
| 37 | \( 1 - 2.89T + 37T^{2} \) |
| 41 | \( 1 + 5.01T + 41T^{2} \) |
| 43 | \( 1 - 1.40T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 1.07T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 9.77T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 9.23T + 79T^{2} \) |
| 83 | \( 1 + 1.78T + 83T^{2} \) |
| 89 | \( 1 + 8.64T + 89T^{2} \) |
| 97 | \( 1 + 5.98T + 97T^{2} \) |
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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
−8.072653099757190461742494153258, −7.37474807610105797219821868542, −6.51255971548691828579463088851, −5.75413536209089853550358924157, −5.04829741643048997763473402008, −4.30028702920232674825224933605, −3.23124823779547509277882140163, −2.86416701139254670662242669819, −1.64305091193420845285226855776, 0,
1.64305091193420845285226855776, 2.86416701139254670662242669819, 3.23124823779547509277882140163, 4.30028702920232674825224933605, 5.04829741643048997763473402008, 5.75413536209089853550358924157, 6.51255971548691828579463088851, 7.37474807610105797219821868542, 8.072653099757190461742494153258