| L(s) = 1 | + 5·5-s − 4.47·7-s + 9·9-s + 11·11-s + 22.3·13-s − 31.3·17-s + 25·25-s − 18·31-s − 22.3·35-s + 84.9·43-s + 45·45-s − 28.9·49-s + 55·55-s + 102·59-s − 40.2·63-s + 111.·65-s + 78·71-s + 4.47·73-s − 49.1·77-s + 81·81-s − 165.·83-s − 156.·85-s + 2·89-s − 100.·91-s + 99·99-s + 156.·107-s + 201.·117-s + ⋯ |
| L(s) = 1 | + 5-s − 0.638·7-s + 9-s + 11-s + 1.72·13-s − 1.84·17-s + 25-s − 0.580·31-s − 0.638·35-s + 1.97·43-s + 45-s − 0.591·49-s + 55-s + 1.72·59-s − 0.638·63-s + 1.72·65-s + 1.09·71-s + 0.0612·73-s − 0.638·77-s + 81-s − 1.99·83-s − 1.84·85-s + 0.0224·89-s − 1.09·91-s + 99-s + 1.46·107-s + 1.72·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.555069892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.555069892\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 9T^{2} \) |
| 7 | \( 1 + 4.47T + 49T^{2} \) |
| 13 | \( 1 - 22.3T + 169T^{2} \) |
| 17 | \( 1 + 31.3T + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 18T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 84.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 102T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 78T + 5.04e3T^{2} \) |
| 73 | \( 1 - 4.47T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 165.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.822216506993771300020655970907, −9.144770480810661761599342856596, −8.581327903440206466450559637991, −7.04609928039003866710160462418, −6.49678412453444965196583785848, −5.82715917604412795329827856680, −4.43686648947965089037969179923, −3.65076653359343296587446965283, −2.17658890556004399258523145112, −1.12080832559685269278955870458,
1.12080832559685269278955870458, 2.17658890556004399258523145112, 3.65076653359343296587446965283, 4.43686648947965089037969179923, 5.82715917604412795329827856680, 6.49678412453444965196583785848, 7.04609928039003866710160462418, 8.581327903440206466450559637991, 9.144770480810661761599342856596, 9.822216506993771300020655970907
