L(s) = 1 | + 2.00·2-s + 0.797·3-s + 2.01·4-s − 2.51·5-s + 1.59·6-s − 4.73·7-s + 0.0356·8-s − 2.36·9-s − 5.03·10-s + 3.14·11-s + 1.61·12-s − 9.49·14-s − 2.00·15-s − 3.96·16-s + 4.16·17-s − 4.73·18-s + 7.56·19-s − 5.06·20-s − 3.77·21-s + 6.29·22-s − 1.78·23-s + 0.0284·24-s + 1.30·25-s − 4.27·27-s − 9.55·28-s − 3.04·29-s − 4.01·30-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.460·3-s + 1.00·4-s − 1.12·5-s + 0.652·6-s − 1.79·7-s + 0.0125·8-s − 0.787·9-s − 1.59·10-s + 0.947·11-s + 0.464·12-s − 2.53·14-s − 0.517·15-s − 0.991·16-s + 1.01·17-s − 1.11·18-s + 1.73·19-s − 1.13·20-s − 0.824·21-s + 1.34·22-s − 0.372·23-s + 0.00579·24-s + 0.261·25-s − 0.823·27-s − 1.80·28-s − 0.565·29-s − 0.733·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.677687490\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.677687490\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - 2.00T + 2T^{2} \) |
| 3 | \( 1 - 0.797T + 3T^{2} \) |
| 5 | \( 1 + 2.51T + 5T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 - 7.56T + 19T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 + 3.04T + 29T^{2} \) |
| 37 | \( 1 - 6.43T + 37T^{2} \) |
| 41 | \( 1 - 7.62T + 41T^{2} \) |
| 43 | \( 1 + 3.32T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 - 6.95T + 59T^{2} \) |
| 61 | \( 1 - 1.86T + 61T^{2} \) |
| 67 | \( 1 + 2.10T + 67T^{2} \) |
| 71 | \( 1 + 4.32T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 4.81T + 79T^{2} \) |
| 83 | \( 1 + 2.04T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 - 4.71T + 97T^{2} \) |
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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
−7.964450913055795923540400930885, −7.36171001489104199468522244478, −6.59735169507520132638220036911, −5.88023336116336130459255423625, −5.39801662735643412071298768748, −4.13112820150324637243848549202, −3.67013197932400844728206201806, −3.23348257840189930657560720396, −2.57843499291675723547785940474, −0.67012775326482862582057574284,
0.67012775326482862582057574284, 2.57843499291675723547785940474, 3.23348257840189930657560720396, 3.67013197932400844728206201806, 4.13112820150324637243848549202, 5.39801662735643412071298768748, 5.88023336116336130459255423625, 6.59735169507520132638220036911, 7.36171001489104199468522244478, 7.964450913055795923540400930885