| L(s) = 1 | + 1.36·2-s − 3·3-s − 6.14·4-s + 17.6·5-s − 4.08·6-s + 31.9·7-s − 19.2·8-s + 9·9-s + 23.9·10-s + 10.4·11-s + 18.4·12-s − 12.2·13-s + 43.5·14-s − 52.8·15-s + 22.9·16-s + 89.2·17-s + 12.2·18-s − 114.·19-s − 108.·20-s − 95.9·21-s + 14.1·22-s − 62.2·23-s + 57.7·24-s + 185.·25-s − 16.6·26-s − 27·27-s − 196.·28-s + ⋯ |
| L(s) = 1 | + 0.481·2-s − 0.577·3-s − 0.768·4-s + 1.57·5-s − 0.277·6-s + 1.72·7-s − 0.850·8-s + 0.333·9-s + 0.758·10-s + 0.285·11-s + 0.443·12-s − 0.260·13-s + 0.831·14-s − 0.910·15-s + 0.359·16-s + 1.27·17-s + 0.160·18-s − 1.37·19-s − 1.21·20-s − 0.997·21-s + 0.137·22-s − 0.564·23-s + 0.491·24-s + 1.48·25-s − 0.125·26-s − 0.192·27-s − 1.32·28-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
| \(L(2)\) |
\(\approx\) |
\(2.740142208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.740142208\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 1.36T + 8T^{2} \) |
| 5 | \( 1 - 17.6T + 125T^{2} \) |
| 7 | \( 1 - 31.9T + 343T^{2} \) |
| 11 | \( 1 - 10.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 89.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 42.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 94.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 506.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 77.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 410.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 439.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 376.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 657.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 23.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 572.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 649.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 579.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 232.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 255.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 893.T + 9.12e5T^{2} \) |
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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
−10.40741313487786017982027624003, −9.955123634242799825832361749811, −8.804264103209302346890503738040, −8.048723265033700358421761860089, −6.53489519800918538394949962620, −5.57549286517690986887842494305, −5.08879568978744696674516504379, −4.11870568631925541929613999086, −2.24067827171464193158375416910, −1.10431511537590611661641412025,
1.10431511537590611661641412025, 2.24067827171464193158375416910, 4.11870568631925541929613999086, 5.08879568978744696674516504379, 5.57549286517690986887842494305, 6.53489519800918538394949962620, 8.048723265033700358421761860089, 8.804264103209302346890503738040, 9.955123634242799825832361749811, 10.40741313487786017982027624003
