| L(s) = 1 | + 2.52·2-s + 4.37·4-s − 3.46·7-s + 5.98·8-s + 1.37·11-s + 4.10·13-s − 8.74·14-s + 6.37·16-s + 2.52·17-s + 5.37·19-s + 3.46·22-s + 5.04·23-s + 10.3·26-s − 15.1·28-s + 5.74·29-s + 0.627·31-s + 4.10·32-s + 6.37·34-s − 7.57·37-s + 13.5·38-s − 1.37·41-s + 3.46·43-s + 6.00·44-s + 12.7·46-s − 8.51·47-s + 4.99·49-s + 17.9·52-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.18·4-s − 1.30·7-s + 2.11·8-s + 0.413·11-s + 1.13·13-s − 2.33·14-s + 1.59·16-s + 0.612·17-s + 1.23·19-s + 0.738·22-s + 1.05·23-s + 2.03·26-s − 2.86·28-s + 1.06·29-s + 0.112·31-s + 0.726·32-s + 1.09·34-s − 1.24·37-s + 2.19·38-s − 0.214·41-s + 0.528·43-s + 0.904·44-s + 1.87·46-s − 1.24·47-s + 0.714·49-s + 2.49·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.197603597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.197603597\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 - 5.04T + 23T^{2} \) |
| 29 | \( 1 - 5.74T + 29T^{2} \) |
| 31 | \( 1 - 0.627T + 31T^{2} \) |
| 37 | \( 1 + 7.57T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 8.51T + 47T^{2} \) |
| 53 | \( 1 + 5.34T + 53T^{2} \) |
| 59 | \( 1 - 7.37T + 59T^{2} \) |
| 61 | \( 1 - 3.62T + 61T^{2} \) |
| 67 | \( 1 - 8.21T + 67T^{2} \) |
| 71 | \( 1 + 4.11T + 71T^{2} \) |
| 73 | \( 1 + 7.57T + 73T^{2} \) |
| 79 | \( 1 + 4.74T + 79T^{2} \) |
| 83 | \( 1 + 5.34T + 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−9.256435419368659133486133844477, −8.220538288747476828253840295120, −6.95888583230421049216157081460, −6.69119635758640159608088865679, −5.79942762852804608199332922616, −5.19963680589248136776790715480, −4.10975677753311819221415088426, −3.31463853360213832684398568366, −2.93093175839376938299028854248, −1.31504907889206821428381877232,
1.31504907889206821428381877232, 2.93093175839376938299028854248, 3.31463853360213832684398568366, 4.10975677753311819221415088426, 5.19963680589248136776790715480, 5.79942762852804608199332922616, 6.69119635758640159608088865679, 6.95888583230421049216157081460, 8.220538288747476828253840295120, 9.256435419368659133486133844477
