L(s) = 1 | − 1.62·3-s − 2.98·5-s − 1.58·7-s − 0.374·9-s + 1.17·11-s − 1.64·13-s + 4.84·15-s − 0.163·17-s − 0.993·19-s + 2.56·21-s − 3.53·23-s + 3.93·25-s + 5.46·27-s − 0.843·29-s − 7.13·31-s − 1.90·33-s + 4.73·35-s + 5.41·37-s + 2.66·39-s − 0.419·41-s − 3.12·43-s + 1.11·45-s − 12.0·47-s − 4.49·49-s + 0.264·51-s − 7.57·53-s − 3.51·55-s + ⋯ |
L(s) = 1 | − 0.935·3-s − 1.33·5-s − 0.598·7-s − 0.124·9-s + 0.354·11-s − 0.455·13-s + 1.25·15-s − 0.0395·17-s − 0.227·19-s + 0.559·21-s − 0.737·23-s + 0.786·25-s + 1.05·27-s − 0.156·29-s − 1.28·31-s − 0.331·33-s + 0.800·35-s + 0.889·37-s + 0.426·39-s − 0.0655·41-s − 0.476·43-s + 0.166·45-s − 1.75·47-s − 0.641·49-s + 0.0370·51-s − 1.04·53-s − 0.474·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05863491482\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05863491482\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 + 1.62T + 3T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 + 0.163T + 17T^{2} \) |
| 19 | \( 1 + 0.993T + 19T^{2} \) |
| 23 | \( 1 + 3.53T + 23T^{2} \) |
| 29 | \( 1 + 0.843T + 29T^{2} \) |
| 31 | \( 1 + 7.13T + 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 + 0.419T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 7.57T + 53T^{2} \) |
| 59 | \( 1 + 6.22T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 1.54T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 + 9.05T + 73T^{2} \) |
| 79 | \( 1 - 5.44T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.81118530839379640330005069497, −7.09067079946422646124643185301, −6.39825917153610352543090194259, −5.88277882875317736571466810387, −4.94302192403671309335639050917, −4.37723328293733095613872103982, −3.55291281882381447585408170533, −2.91317641498704423188178567052, −1.56437242989173278639285277309, −0.12503113982004682494752560533,
0.12503113982004682494752560533, 1.56437242989173278639285277309, 2.91317641498704423188178567052, 3.55291281882381447585408170533, 4.37723328293733095613872103982, 4.94302192403671309335639050917, 5.88277882875317736571466810387, 6.39825917153610352543090194259, 7.09067079946422646124643185301, 7.81118530839379640330005069497