| L(s) = 1 | − 3.59e3·3-s + 1.95e6·5-s − 1.23e8·7-s − 1.14e9·9-s + 1.18e9·11-s + 3.28e10·13-s − 7.01e9·15-s − 6.85e11·17-s + 1.89e12·19-s + 4.43e11·21-s − 1.26e12·23-s + 3.81e12·25-s + 8.30e12·27-s + 1.34e14·29-s + 2.65e14·31-s − 4.25e12·33-s − 2.41e14·35-s − 8.78e14·37-s − 1.18e14·39-s + 1.34e15·41-s − 2.93e15·43-s − 2.24e15·45-s + 2.27e15·47-s + 3.83e15·49-s + 2.46e15·51-s − 3.83e16·53-s + 2.31e15·55-s + ⋯ |
| L(s) = 1 | − 0.105·3-s + 0.447·5-s − 1.15·7-s − 0.988·9-s + 0.151·11-s + 0.859·13-s − 0.0471·15-s − 1.40·17-s + 1.34·19-s + 0.121·21-s − 0.146·23-s + 0.199·25-s + 0.209·27-s + 1.72·29-s + 1.80·31-s − 0.0159·33-s − 0.516·35-s − 1.11·37-s − 0.0905·39-s + 0.643·41-s − 0.890·43-s − 0.442·45-s + 0.297·47-s + 0.336·49-s + 0.147·51-s − 1.59·53-s + 0.0677·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 1.95e6T \) |
| good | 3 | \( 1 + 3.59e3T + 1.16e9T^{2} \) |
| 7 | \( 1 + 1.23e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.18e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 3.28e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 6.85e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.89e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.26e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 1.34e14T + 6.10e27T^{2} \) |
| 31 | \( 1 - 2.65e14T + 2.16e28T^{2} \) |
| 37 | \( 1 + 8.78e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 1.34e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 2.93e15T + 1.08e31T^{2} \) |
| 47 | \( 1 - 2.27e15T + 5.88e31T^{2} \) |
| 53 | \( 1 + 3.83e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 8.16e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.01e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 1.40e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 5.34e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 4.29e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 6.87e16T + 1.13e36T^{2} \) |
| 83 | \( 1 - 2.54e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.68e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 3.86e18T + 5.60e37T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.20031267183815250001188000469, −9.209653764962290024668446841470, −8.316053200702545218170616217471, −6.63989641719973587571570074125, −6.13611781493570646055781802893, −4.84287714724889842182861314855, −3.37183598717936535584800463256, −2.59837567016085317445389111376, −1.09553920059368461930950567558, 0,
1.09553920059368461930950567558, 2.59837567016085317445389111376, 3.37183598717936535584800463256, 4.84287714724889842182861314855, 6.13611781493570646055781802893, 6.63989641719973587571570074125, 8.316053200702545218170616217471, 9.209653764962290024668446841470, 10.20031267183815250001188000469
