| L(s) = 1 | − 1.90·3-s + 3.78·5-s + 0.441·7-s + 0.622·9-s + 3.73·11-s + 0.731·13-s − 7.19·15-s − 2.10·17-s − 7.92·19-s − 0.839·21-s − 7.60·23-s + 9.30·25-s + 4.52·27-s + 4.11·29-s − 4.88·31-s − 7.10·33-s + 1.66·35-s − 10.6·37-s − 1.39·39-s − 9.14·41-s − 4.80·43-s + 2.35·45-s − 5.33·47-s − 6.80·49-s + 4.00·51-s − 9.64·53-s + 14.1·55-s + ⋯ |
| L(s) = 1 | − 1.09·3-s + 1.69·5-s + 0.166·7-s + 0.207·9-s + 1.12·11-s + 0.202·13-s − 1.85·15-s − 0.510·17-s − 1.81·19-s − 0.183·21-s − 1.58·23-s + 1.86·25-s + 0.870·27-s + 0.764·29-s − 0.876·31-s − 1.23·33-s + 0.282·35-s − 1.74·37-s − 0.223·39-s − 1.42·41-s − 0.732·43-s + 0.351·45-s − 0.778·47-s − 0.972·49-s + 0.561·51-s − 1.32·53-s + 1.90·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 1.90T + 3T^{2} \) |
| 5 | \( 1 - 3.78T + 5T^{2} \) |
| 7 | \( 1 - 0.441T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 - 0.731T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 23 | \( 1 + 7.60T + 23T^{2} \) |
| 29 | \( 1 - 4.11T + 29T^{2} \) |
| 31 | \( 1 + 4.88T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 9.14T + 41T^{2} \) |
| 43 | \( 1 + 4.80T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 + 9.64T + 53T^{2} \) |
| 59 | \( 1 + 6.91T + 59T^{2} \) |
| 61 | \( 1 - 1.47T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 - 0.0833T + 73T^{2} \) |
| 79 | \( 1 + 9.43T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 2.63T + 97T^{2} \) |
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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
−8.397720253716447722059553197580, −6.78922195353401488380192604568, −6.45396534509159456714765458412, −6.04093762182746350889051068698, −5.19420147208366688695138981815, −4.57035439092497940607887155024, −3.46129870432223995343255881900, −2.03735645034927012588691436541, −1.61393202819819530068506581777, 0,
1.61393202819819530068506581777, 2.03735645034927012588691436541, 3.46129870432223995343255881900, 4.57035439092497940607887155024, 5.19420147208366688695138981815, 6.04093762182746350889051068698, 6.45396534509159456714765458412, 6.78922195353401488380192604568, 8.397720253716447722059553197580
