L(s) = 1 | + 2.78·3-s + 5-s + 0.680·7-s + 4.74·9-s − 3.41·11-s − 7.11·13-s + 2.78·15-s + 1.64·17-s − 3.37·19-s + 1.89·21-s − 5.06·23-s + 25-s + 4.85·27-s − 6.60·29-s − 7.95·31-s − 9.50·33-s + 0.680·35-s + 3.46·37-s − 19.8·39-s + 5.62·41-s + 12.4·43-s + 4.74·45-s − 0.196·47-s − 6.53·49-s + 4.57·51-s − 4.67·53-s − 3.41·55-s + ⋯ |
L(s) = 1 | + 1.60·3-s + 0.447·5-s + 0.257·7-s + 1.58·9-s − 1.03·11-s − 1.97·13-s + 0.718·15-s + 0.398·17-s − 0.774·19-s + 0.413·21-s − 1.05·23-s + 0.200·25-s + 0.934·27-s − 1.22·29-s − 1.42·31-s − 1.65·33-s + 0.115·35-s + 0.569·37-s − 3.17·39-s + 0.878·41-s + 1.90·43-s + 0.707·45-s − 0.0286·47-s − 0.933·49-s + 0.639·51-s − 0.641·53-s − 0.460·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 2.78T + 3T^{2} \) |
| 7 | \( 1 - 0.680T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 7.11T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 + 6.60T + 29T^{2} \) |
| 31 | \( 1 + 7.95T + 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 - 5.62T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 0.196T + 47T^{2} \) |
| 53 | \( 1 + 4.67T + 53T^{2} \) |
| 59 | \( 1 + 3.58T + 59T^{2} \) |
| 61 | \( 1 - 2.00T + 61T^{2} \) |
| 67 | \( 1 - 0.548T + 67T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 + 4.68T + 73T^{2} \) |
| 79 | \( 1 + 8.19T + 79T^{2} \) |
| 83 | \( 1 - 4.22T + 83T^{2} \) |
| 89 | \( 1 + 2.89T + 89T^{2} \) |
| 97 | \( 1 + 1.63T + 97T^{2} \) |
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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
−7.51891493553928063202082568635, −7.31738316023767417218941831435, −6.04372252956796369503731186719, −5.33498957924361031457757476693, −4.52957396402513180169327364246, −3.82785864793922414353832449313, −2.81694208853800779484071345092, −2.34915889166515838457617612119, −1.77516149606755108416228100070, 0,
1.77516149606755108416228100070, 2.34915889166515838457617612119, 2.81694208853800779484071345092, 3.82785864793922414353832449313, 4.52957396402513180169327364246, 5.33498957924361031457757476693, 6.04372252956796369503731186719, 7.31738316023767417218941831435, 7.51891493553928063202082568635