| L(s) = 1 | + 40.5·2-s + 81·3-s + 1.13e3·4-s − 281.·5-s + 3.28e3·6-s − 1.19e4·7-s + 2.52e4·8-s + 6.56e3·9-s − 1.14e4·10-s + 3.80e4·11-s + 9.18e4·12-s + 1.24e5·13-s − 4.84e5·14-s − 2.28e4·15-s + 4.42e5·16-s + 4.83e5·17-s + 2.66e5·18-s + 1.18e5·19-s − 3.19e5·20-s − 9.67e5·21-s + 1.54e6·22-s + 1.81e6·23-s + 2.04e6·24-s − 1.87e6·25-s + 5.05e6·26-s + 5.31e5·27-s − 1.35e7·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 0.577·3-s + 2.21·4-s − 0.201·5-s + 1.03·6-s − 1.88·7-s + 2.17·8-s + 0.333·9-s − 0.361·10-s + 0.783·11-s + 1.27·12-s + 1.20·13-s − 3.37·14-s − 0.116·15-s + 1.68·16-s + 1.40·17-s + 0.597·18-s + 0.209·19-s − 0.446·20-s − 1.08·21-s + 1.40·22-s + 1.34·23-s + 1.25·24-s − 0.959·25-s + 2.16·26-s + 0.192·27-s − 4.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(8.251927318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.251927318\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 - 40.5T + 512T^{2} \) |
| 5 | \( 1 + 281.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.19e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.80e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.24e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.83e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.18e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.81e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.59e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.75e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.86e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.23e5T + 3.27e14T^{2} \) |
| 43 | \( 1 - 9.66e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.98e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.40e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 1.41e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.24e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.18e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.38e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.97e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.52e6T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.14e9T + 7.60e17T^{2} \) |
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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
−11.44738011647106546390022825528, −10.14102066397760599782350877173, −9.109913332848081315030864651980, −7.50603349236435317265142335267, −6.43609020316761290556106025479, −5.86998600940315692617681686384, −4.25233656858561225890602857030, −3.38365616294759084151228550882, −2.90682104372935137941585478532, −1.13235026194029287114534943108,
1.13235026194029287114534943108, 2.90682104372935137941585478532, 3.38365616294759084151228550882, 4.25233656858561225890602857030, 5.86998600940315692617681686384, 6.43609020316761290556106025479, 7.50603349236435317265142335267, 9.109913332848081315030864651980, 10.14102066397760599782350877173, 11.44738011647106546390022825528
