| L(s) = 1 | − 0.551·3-s + 0.616·5-s + 1.02·7-s − 2.69·9-s + 6.13·11-s − 4.21·13-s − 0.340·15-s + 17-s + 7.73·19-s − 0.564·21-s + 0.222·23-s − 4.61·25-s + 3.14·27-s − 3.60·29-s + 8.79·31-s − 3.38·33-s + 0.631·35-s + 11.6·37-s + 2.32·39-s − 6.47·41-s + 7.10·43-s − 1.66·45-s − 10.8·47-s − 5.95·49-s − 0.551·51-s + 4.52·53-s + 3.78·55-s + ⋯ |
| L(s) = 1 | − 0.318·3-s + 0.275·5-s + 0.386·7-s − 0.898·9-s + 1.84·11-s − 1.16·13-s − 0.0877·15-s + 0.242·17-s + 1.77·19-s − 0.123·21-s + 0.0464·23-s − 0.923·25-s + 0.604·27-s − 0.669·29-s + 1.57·31-s − 0.588·33-s + 0.106·35-s + 1.91·37-s + 0.371·39-s − 1.01·41-s + 1.08·43-s − 0.247·45-s − 1.57·47-s − 0.850·49-s − 0.0772·51-s + 0.621·53-s + 0.509·55-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
| \(L(1)\) |
\(\approx\) |
\(2.094778407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.094778407\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.551T + 3T^{2} \) |
| 5 | \( 1 - 0.616T + 5T^{2} \) |
| 7 | \( 1 - 1.02T + 7T^{2} \) |
| 11 | \( 1 - 6.13T + 11T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 19 | \( 1 - 7.73T + 19T^{2} \) |
| 23 | \( 1 - 0.222T + 23T^{2} \) |
| 29 | \( 1 + 3.60T + 29T^{2} \) |
| 31 | \( 1 - 8.79T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 - 7.10T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 4.52T + 53T^{2} \) |
| 61 | \( 1 - 0.968T + 61T^{2} \) |
| 67 | \( 1 + 4.46T + 67T^{2} \) |
| 71 | \( 1 - 4.36T + 71T^{2} \) |
| 73 | \( 1 + 9.72T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 8.32T + 83T^{2} \) |
| 89 | \( 1 + 4.83T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.78266462926824613734624467340, −7.15398688550486401508502737617, −6.30696389857925081833164628607, −5.84999732416624848929166448529, −5.04258246748565045600795444071, −4.42018465681261068972127249082, −3.45834807996363625664098574889, −2.71546169573062092270067532377, −1.65754065783025870030300188669, −0.76090703284800193688308519646,
0.76090703284800193688308519646, 1.65754065783025870030300188669, 2.71546169573062092270067532377, 3.45834807996363625664098574889, 4.42018465681261068972127249082, 5.04258246748565045600795444071, 5.84999732416624848929166448529, 6.30696389857925081833164628607, 7.15398688550486401508502737617, 7.78266462926824613734624467340
