L(s) = 1 | + 5.26·2-s − 3·3-s + 19.7·4-s + 11.8·5-s − 15.7·6-s − 3.36·7-s + 61.6·8-s + 9·9-s + 62.5·10-s − 3.40·11-s − 59.1·12-s − 54.8·13-s − 17.7·14-s − 35.6·15-s + 166.·16-s + 68.7·17-s + 47.3·18-s + 8.88·19-s + 234.·20-s + 10.0·21-s − 17.9·22-s + 80.8·23-s − 184.·24-s + 16.1·25-s − 288.·26-s − 27·27-s − 66.3·28-s + ⋯ |
L(s) = 1 | + 1.86·2-s − 0.577·3-s + 2.46·4-s + 1.06·5-s − 1.07·6-s − 0.181·7-s + 2.72·8-s + 0.333·9-s + 1.97·10-s − 0.0932·11-s − 1.42·12-s − 1.17·13-s − 0.338·14-s − 0.613·15-s + 2.60·16-s + 0.980·17-s + 0.620·18-s + 0.107·19-s + 2.61·20-s + 0.104·21-s − 0.173·22-s + 0.732·23-s − 1.57·24-s + 0.129·25-s − 2.17·26-s − 0.192·27-s − 0.447·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.857878983\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.857878983\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 59 | \( 1 + 59T \) |
good | 2 | \( 1 - 5.26T + 8T^{2} \) |
| 5 | \( 1 - 11.8T + 125T^{2} \) |
| 7 | \( 1 + 3.36T + 343T^{2} \) |
| 11 | \( 1 + 3.40T + 1.33e3T^{2} \) |
| 13 | \( 1 + 54.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 8.88T + 6.85e3T^{2} \) |
| 23 | \( 1 - 80.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 145.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 37.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 465.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 271.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 82.0T + 1.48e5T^{2} \) |
| 61 | \( 1 + 736.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 768.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 164.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 445.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 602.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 779.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 269.T + 9.12e5T^{2} \) |
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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
−12.43642218462910619726380330768, −11.67225244959342003165816301135, −10.53590677684481881085395359166, −9.652020898345426902958275931313, −7.47246303112744110486387561006, −6.48551601027429180692333594700, −5.52150165195575619991365254423, −4.89028542887256936587192330219, −3.30402882192459559539321681417, −1.92412853782076450916311592169,
1.92412853782076450916311592169, 3.30402882192459559539321681417, 4.89028542887256936587192330219, 5.52150165195575619991365254423, 6.48551601027429180692333594700, 7.47246303112744110486387561006, 9.652020898345426902958275931313, 10.53590677684481881085395359166, 11.67225244959342003165816301135, 12.43642218462910619726380330768