L(s) = 1 | − 2-s − 1.41·3-s + 4-s − 3.55·5-s + 1.41·6-s − 3.54·7-s − 8-s − 1.00·9-s + 3.55·10-s + 4.87·11-s − 1.41·12-s − 3.10·13-s + 3.54·14-s + 5.00·15-s + 16-s − 4.40·17-s + 1.00·18-s + 1.30·19-s − 3.55·20-s + 5.00·21-s − 4.87·22-s − 3.42·23-s + 1.41·24-s + 7.60·25-s + 3.10·26-s + 5.65·27-s − 3.54·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.814·3-s + 0.5·4-s − 1.58·5-s + 0.576·6-s − 1.34·7-s − 0.353·8-s − 0.336·9-s + 1.12·10-s + 1.46·11-s − 0.407·12-s − 0.861·13-s + 0.948·14-s + 1.29·15-s + 0.250·16-s − 1.06·17-s + 0.237·18-s + 0.299·19-s − 0.793·20-s + 1.09·21-s − 1.03·22-s − 0.713·23-s + 0.288·24-s + 1.52·25-s + 0.609·26-s + 1.08·27-s − 0.670·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 13 | \( 1 + 3.10T + 13T^{2} \) |
| 17 | \( 1 + 4.40T + 17T^{2} \) |
| 19 | \( 1 - 1.30T + 19T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 - 8.36T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 8.60T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 + 6.84T + 83T^{2} \) |
| 89 | \( 1 + 2.84T + 89T^{2} \) |
| 97 | \( 1 - 0.0168T + 97T^{2} \) |
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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
−8.333415807601977326155997183546, −7.08194767280800414547314238895, −6.77504333166065072206743368464, −6.23132882527228251618015607654, −5.08087201687663732776694025931, −4.12637485274196318309568574880, −3.48476089610824896840666954413, −2.51307324821246515238699623249, −0.792574291436769231333454711578, 0,
0.792574291436769231333454711578, 2.51307324821246515238699623249, 3.48476089610824896840666954413, 4.12637485274196318309568574880, 5.08087201687663732776694025931, 6.23132882527228251618015607654, 6.77504333166065072206743368464, 7.08194767280800414547314238895, 8.333415807601977326155997183546