L(s) = 1 | − 1.80·3-s + 3.22·5-s + 2.17·7-s + 0.255·9-s − 3.64·11-s − 6.01·13-s − 5.81·15-s + 17-s − 0.114·19-s − 3.92·21-s + 1.90·23-s + 5.38·25-s + 4.95·27-s − 6.34·29-s − 2.24·31-s + 6.57·33-s + 7.00·35-s + 0.419·37-s + 10.8·39-s + 6.73·41-s + 12.4·43-s + 0.824·45-s + 6.03·47-s − 2.27·49-s − 1.80·51-s − 1.79·53-s − 11.7·55-s + ⋯ |
L(s) = 1 | − 1.04·3-s + 1.44·5-s + 0.821·7-s + 0.0852·9-s − 1.09·11-s − 1.66·13-s − 1.50·15-s + 0.242·17-s − 0.0262·19-s − 0.855·21-s + 0.397·23-s + 1.07·25-s + 0.952·27-s − 1.17·29-s − 0.402·31-s + 1.14·33-s + 1.18·35-s + 0.0688·37-s + 1.73·39-s + 1.05·41-s + 1.89·43-s + 0.122·45-s + 0.880·47-s − 0.325·49-s − 0.252·51-s − 0.247·53-s − 1.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 1.80T + 3T^{2} \) |
| 5 | \( 1 - 3.22T + 5T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 + 6.01T + 13T^{2} \) |
| 19 | \( 1 + 0.114T + 19T^{2} \) |
| 23 | \( 1 - 1.90T + 23T^{2} \) |
| 29 | \( 1 + 6.34T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 0.419T + 37T^{2} \) |
| 41 | \( 1 - 6.73T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 6.03T + 47T^{2} \) |
| 53 | \( 1 + 1.79T + 53T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 1.45T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 6.48T + 79T^{2} \) |
| 83 | \( 1 - 5.58T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−7.45416249521445977203602470616, −6.71146490860832224130651043808, −5.78090350429471244866319831863, −5.41610309621300769545195340460, −5.08705496990727123473649453172, −4.21413360340549561248375538198, −2.64151561278240486969974144841, −2.34697283336519728891780347350, −1.22874045859818616702458760091, 0,
1.22874045859818616702458760091, 2.34697283336519728891780347350, 2.64151561278240486969974144841, 4.21413360340549561248375538198, 5.08705496990727123473649453172, 5.41610309621300769545195340460, 5.78090350429471244866319831863, 6.71146490860832224130651043808, 7.45416249521445977203602470616