| L(s) = 1 | + 3.23·3-s + 3.23·5-s − 7-s + 7.47·9-s − 11-s + 1.23·13-s + 10.4·15-s − 6.47·17-s + 2.76·19-s − 3.23·21-s − 4·23-s + 5.47·25-s + 14.4·27-s − 4.47·29-s − 2·31-s − 3.23·33-s − 3.23·35-s − 10.9·37-s + 4.00·39-s + 6.47·41-s + 1.52·43-s + 24.1·45-s + 2·47-s + 49-s − 20.9·51-s − 0.472·53-s − 3.23·55-s + ⋯ |
| L(s) = 1 | + 1.86·3-s + 1.44·5-s − 0.377·7-s + 2.49·9-s − 0.301·11-s + 0.342·13-s + 2.70·15-s − 1.56·17-s + 0.634·19-s − 0.706·21-s − 0.834·23-s + 1.09·25-s + 2.78·27-s − 0.830·29-s − 0.359·31-s − 0.563·33-s − 0.546·35-s − 1.79·37-s + 0.640·39-s + 1.01·41-s + 0.232·43-s + 3.60·45-s + 0.291·47-s + 0.142·49-s − 2.93·51-s − 0.0648·53-s − 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.771272727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.771272727\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 4.94T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−9.432241291593031985312755966401, −9.090867750183411622192458206098, −8.305273934807586744371261108078, −7.33688902175508696642818110586, −6.56337477995572188610046899863, −5.54271814798355277903266168272, −4.30531465936670052491121869970, −3.31958618468901417052720891250, −2.34663910053050906955457158406, −1.74813798165055337946007207421,
1.74813798165055337946007207421, 2.34663910053050906955457158406, 3.31958618468901417052720891250, 4.30531465936670052491121869970, 5.54271814798355277903266168272, 6.56337477995572188610046899863, 7.33688902175508696642818110586, 8.305273934807586744371261108078, 9.090867750183411622192458206098, 9.432241291593031985312755966401
