| L(s) = 1 | + 4.89·5-s + 10.6·7-s + 35.4·11-s + 72.7·13-s + 127.·17-s + 46.3·19-s − 131.·23-s − 101.·25-s + 137.·29-s + 106.·31-s + 52.1·35-s + 137.·37-s + 71.7·41-s − 376.·43-s − 613.·47-s − 229.·49-s + 431.·53-s + 173.·55-s − 285.·59-s − 43.9·61-s + 356.·65-s + 45.2·67-s + 357.·71-s + 530.·73-s + 377.·77-s + 195.·79-s + 760.·83-s + ⋯ |
| L(s) = 1 | + 0.438·5-s + 0.575·7-s + 0.971·11-s + 1.55·13-s + 1.81·17-s + 0.560·19-s − 1.18·23-s − 0.808·25-s + 0.880·29-s + 0.614·31-s + 0.252·35-s + 0.610·37-s + 0.273·41-s − 1.33·43-s − 1.90·47-s − 0.669·49-s + 1.11·53-s + 0.425·55-s − 0.630·59-s − 0.0922·61-s + 0.679·65-s + 0.0824·67-s + 0.597·71-s + 0.850·73-s + 0.559·77-s + 0.277·79-s + 1.00·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.279091041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.279091041\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.89T + 125T^{2} \) |
| 7 | \( 1 - 10.6T + 343T^{2} \) |
| 11 | \( 1 - 35.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 46.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 106.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 71.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 376.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 613.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 431.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 285.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 43.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 45.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 357.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 530.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 195.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 760.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3T + 9.12e5T^{2} \) |
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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
−9.448187302897978856758724704730, −8.202927017747457390228519641928, −8.014306870558987899949254287220, −6.60410626462224019394554522871, −6.02127840989425366086275783721, −5.16760572171024869458821582871, −3.99625204779079031634013330256, −3.25118129478598376514948025829, −1.71977778992074275282340516548, −1.02954980424000490271891367329,
1.02954980424000490271891367329, 1.71977778992074275282340516548, 3.25118129478598376514948025829, 3.99625204779079031634013330256, 5.16760572171024869458821582871, 6.02127840989425366086275783721, 6.60410626462224019394554522871, 8.014306870558987899949254287220, 8.202927017747457390228519641928, 9.448187302897978856758724704730
