| L(s) = 1 | + 1.86·2-s − 3-s + 1.47·4-s + 4.02·5-s − 1.86·6-s + 3.22·7-s − 0.986·8-s + 9-s + 7.50·10-s − 3.00·11-s − 1.47·12-s − 13-s + 6.00·14-s − 4.02·15-s − 4.77·16-s + 4.28·17-s + 1.86·18-s − 6.27·19-s + 5.92·20-s − 3.22·21-s − 5.59·22-s + 7.70·23-s + 0.986·24-s + 11.2·25-s − 1.86·26-s − 27-s + 4.74·28-s + ⋯ |
| L(s) = 1 | + 1.31·2-s − 0.577·3-s + 0.735·4-s + 1.80·5-s − 0.760·6-s + 1.21·7-s − 0.348·8-s + 0.333·9-s + 2.37·10-s − 0.905·11-s − 0.424·12-s − 0.277·13-s + 1.60·14-s − 1.04·15-s − 1.19·16-s + 1.03·17-s + 0.439·18-s − 1.43·19-s + 1.32·20-s − 0.703·21-s − 1.19·22-s + 1.60·23-s + 0.201·24-s + 2.24·25-s − 0.365·26-s − 0.192·27-s + 0.896·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.830408163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.830408163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.86T + 2T^{2} \) |
| 5 | \( 1 - 4.02T + 5T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 19 | \( 1 + 6.27T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 + 5.15T + 29T^{2} \) |
| 31 | \( 1 - 6.62T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 - 1.36T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 5.13T + 47T^{2} \) |
| 53 | \( 1 + 0.715T + 53T^{2} \) |
| 59 | \( 1 - 4.67T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 + 1.94T + 67T^{2} \) |
| 71 | \( 1 - 8.42T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 4.74T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 3.76T + 89T^{2} \) |
| 97 | \( 1 + 8.35T + 97T^{2} \) |
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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
−8.454282745153125217157095587275, −7.46745081543172146425123605902, −6.59567247768395259773983480040, −5.80366289240823944644499500141, −5.49213213403008172498557034085, −4.85822343467650420061535973552, −4.25911511188823329633137991995, −2.78340602591298142622283710023, −2.28297797951369606158149276670, −1.14753019196009128432935547720,
1.14753019196009128432935547720, 2.28297797951369606158149276670, 2.78340602591298142622283710023, 4.25911511188823329633137991995, 4.85822343467650420061535973552, 5.49213213403008172498557034085, 5.80366289240823944644499500141, 6.59567247768395259773983480040, 7.46745081543172146425123605902, 8.454282745153125217157095587275
