| L(s) = 1 | + 15.4·5-s − 23.8·7-s + 14.2·11-s + 13.8·13-s − 80.5·17-s − 144.·19-s + 141.·23-s + 112.·25-s + 251.·29-s + 16.6·31-s − 367.·35-s − 305.·37-s + 429.·41-s − 181.·43-s + 79.4·47-s + 225.·49-s + 663.·53-s + 219.·55-s + 220.·59-s + 473.·61-s + 213.·65-s − 647.·67-s − 14.4·71-s + 776.·73-s − 339.·77-s + 257.·79-s + 1.28e3·83-s + ⋯ |
| L(s) = 1 | + 1.37·5-s − 1.28·7-s + 0.390·11-s + 0.295·13-s − 1.14·17-s − 1.74·19-s + 1.27·23-s + 0.901·25-s + 1.60·29-s + 0.0965·31-s − 1.77·35-s − 1.35·37-s + 1.63·41-s − 0.644·43-s + 0.246·47-s + 0.655·49-s + 1.72·53-s + 0.538·55-s + 0.486·59-s + 0.993·61-s + 0.406·65-s − 1.18·67-s − 0.0242·71-s + 1.24·73-s − 0.502·77-s + 0.367·79-s + 1.69·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.294738608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.294738608\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 15.4T + 125T^{2} \) |
| 7 | \( 1 + 23.8T + 343T^{2} \) |
| 11 | \( 1 - 14.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 141.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 251.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 16.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 305.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 79.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 663.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 220.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 473.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 647.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 14.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 776.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 257.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 156.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
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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
−8.961052865197363641262310979800, −8.570124414095437227670841300155, −6.93095979027148754673578902114, −6.54122648357259722620947717799, −5.97572279147423838195351248653, −4.91920545884727295483239446811, −3.91128900112937057346789878178, −2.76479877754158575867188921355, −2.04416846613523531582556795924, −0.70392372668962425286367885466,
0.70392372668962425286367885466, 2.04416846613523531582556795924, 2.76479877754158575867188921355, 3.91128900112937057346789878178, 4.91920545884727295483239446811, 5.97572279147423838195351248653, 6.54122648357259722620947717799, 6.93095979027148754673578902114, 8.570124414095437227670841300155, 8.961052865197363641262310979800
