| L(s) = 1 | − 2.32·2-s − 3.21·3-s + 3.42·4-s − 3.16·5-s + 7.49·6-s − 7-s − 3.32·8-s + 7.34·9-s + 7.38·10-s + 0.406·11-s − 11.0·12-s + 3.07·13-s + 2.32·14-s + 10.1·15-s + 0.886·16-s + 17-s − 17.1·18-s − 1.57·19-s − 10.8·20-s + 3.21·21-s − 0.946·22-s + 0.852·23-s + 10.6·24-s + 5.04·25-s − 7.17·26-s − 13.9·27-s − 3.42·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 1.85·3-s + 1.71·4-s − 1.41·5-s + 3.05·6-s − 0.377·7-s − 1.17·8-s + 2.44·9-s + 2.33·10-s + 0.122·11-s − 3.18·12-s + 0.854·13-s + 0.622·14-s + 2.63·15-s + 0.221·16-s + 0.242·17-s − 4.03·18-s − 0.362·19-s − 2.42·20-s + 0.701·21-s − 0.201·22-s + 0.177·23-s + 2.18·24-s + 1.00·25-s − 1.40·26-s − 2.68·27-s − 0.647·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1853277118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1853277118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 3 | \( 1 + 3.21T + 3T^{2} \) |
| 5 | \( 1 + 3.16T + 5T^{2} \) |
| 11 | \( 1 - 0.406T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 19 | \( 1 + 1.57T + 19T^{2} \) |
| 23 | \( 1 - 0.852T + 23T^{2} \) |
| 29 | \( 1 + 1.49T + 29T^{2} \) |
| 31 | \( 1 + 0.316T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 - 9.09T + 47T^{2} \) |
| 53 | \( 1 + 4.34T + 53T^{2} \) |
| 59 | \( 1 - 7.15T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 - 5.48T + 67T^{2} \) |
| 71 | \( 1 + 8.73T + 71T^{2} \) |
| 73 | \( 1 - 6.25T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 + 5.35T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 0.623T + 97T^{2} \) |
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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
−12.88816592474920485272715543053, −11.87368465780172161216114870837, −11.18718241203507910059826019837, −10.62429551014660040152677537438, −9.408183628649843887268132383376, −7.997382067082151420265125893493, −7.08619848431718227397248397586, −6.05118858654806093958592741880, −4.22456987191560829905691626011, −0.75686502916089684582990509027,
0.75686502916089684582990509027, 4.22456987191560829905691626011, 6.05118858654806093958592741880, 7.08619848431718227397248397586, 7.997382067082151420265125893493, 9.408183628649843887268132383376, 10.62429551014660040152677537438, 11.18718241203507910059826019837, 11.87368465780172161216114870837, 12.88816592474920485272715543053
