| L(s) = 1 | − 1.38·2-s − 3·3-s − 6.09·4-s + 20.1·5-s + 4.14·6-s − 9.60·7-s + 19.4·8-s + 9·9-s − 27.9·10-s − 17.8·11-s + 18.2·12-s + 69.1·13-s + 13.2·14-s − 60.5·15-s + 21.8·16-s + 0.226·17-s − 12.4·18-s − 28.7·19-s − 123.·20-s + 28.8·21-s + 24.6·22-s − 45.3·23-s − 58.4·24-s + 282.·25-s − 95.4·26-s − 27·27-s + 58.4·28-s + ⋯ |
| L(s) = 1 | − 0.488·2-s − 0.577·3-s − 0.761·4-s + 1.80·5-s + 0.282·6-s − 0.518·7-s + 0.860·8-s + 0.333·9-s − 0.882·10-s − 0.488·11-s + 0.439·12-s + 1.47·13-s + 0.253·14-s − 1.04·15-s + 0.340·16-s + 0.00323·17-s − 0.162·18-s − 0.347·19-s − 1.37·20-s + 0.299·21-s + 0.238·22-s − 0.410·23-s − 0.496·24-s + 2.26·25-s − 0.720·26-s − 0.192·27-s + 0.394·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.345473019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345473019\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 157 | \( 1 + 157T \) |
| good | 2 | \( 1 + 1.38T + 8T^{2} \) |
| 5 | \( 1 - 20.1T + 125T^{2} \) |
| 7 | \( 1 + 9.60T + 343T^{2} \) |
| 11 | \( 1 + 17.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 69.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 0.226T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 93.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 45.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 305.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 78.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 32.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 54.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 785.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 715.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 885.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 951.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 689.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 896.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−10.48857135347388108763768910210, −9.624143859811988686343072203955, −9.141893022901527972563380063203, −8.096756923801759369994554677212, −6.67563157827934908975241168555, −5.85781296302696827737401649189, −5.20041461661277681931505414708, −3.78979228850647557289251012626, −2.05691145503684518461386759293, −0.842438272844092185242068646376,
0.842438272844092185242068646376, 2.05691145503684518461386759293, 3.78979228850647557289251012626, 5.20041461661277681931505414708, 5.85781296302696827737401649189, 6.67563157827934908975241168555, 8.096756923801759369994554677212, 9.141893022901527972563380063203, 9.624143859811988686343072203955, 10.48857135347388108763768910210
