| L(s) = 1 | + 42.2·2-s − 17.8·3-s + 1.27e3·4-s − 756.·6-s + 1.01e4·7-s + 3.23e4·8-s − 1.93e4·9-s + 3.58e4·11-s − 2.28e4·12-s + 2.85e4·13-s + 4.30e5·14-s + 7.14e5·16-s − 3.03e5·17-s − 8.18e5·18-s + 1.23e5·19-s − 1.81e5·21-s + 1.51e6·22-s + 2.12e6·23-s − 5.78e5·24-s + 1.20e6·26-s + 6.97e5·27-s + 1.29e7·28-s − 3.35e6·29-s + 6.59e6·31-s + 1.36e7·32-s − 6.40e5·33-s − 1.28e7·34-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.127·3-s + 2.49·4-s − 0.238·6-s + 1.60·7-s + 2.79·8-s − 0.983·9-s + 0.738·11-s − 0.317·12-s + 0.277·13-s + 2.99·14-s + 2.72·16-s − 0.880·17-s − 1.83·18-s + 0.217·19-s − 0.203·21-s + 1.38·22-s + 1.58·23-s − 0.355·24-s + 0.518·26-s + 0.252·27-s + 3.99·28-s − 0.880·29-s + 1.28·31-s + 2.30·32-s − 0.0940·33-s − 1.64·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(10.35212475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.35212475\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 - 42.2T + 512T^{2} \) |
| 3 | \( 1 + 17.8T + 1.96e4T^{2} \) |
| 7 | \( 1 - 1.01e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.58e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 3.03e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.23e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.35e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.59e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.47e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.71e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.47e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.96e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.53e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.08e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.53e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.35e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.27e6T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.91e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.06e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.84e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.16e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.54e7T + 7.60e17T^{2} \) |
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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
−10.78121755288456941843975222050, −8.948072165892837623541159049024, −7.899944193533066665254474859118, −6.80951272472363780776566800438, −5.90223767164271507686231167114, −4.98486379328687451303956696056, −4.40002674403141259689194591415, −3.20810557010890058730384644547, −2.18842414934793236671555908489, −1.15326545154654773331422683493,
1.15326545154654773331422683493, 2.18842414934793236671555908489, 3.20810557010890058730384644547, 4.40002674403141259689194591415, 4.98486379328687451303956696056, 5.90223767164271507686231167114, 6.80951272472363780776566800438, 7.899944193533066665254474859118, 8.948072165892837623541159049024, 10.78121755288456941843975222050
