| L(s) = 1 | + 9.62·2-s − 419.·4-s − 636.·5-s − 3.28e3·7-s − 8.96e3·8-s − 6.12e3·10-s + 7.18e4·11-s − 5.64e4·13-s − 3.16e4·14-s + 1.28e5·16-s − 2.27e5·17-s − 2.58e5·19-s + 2.66e5·20-s + 6.92e5·22-s + 2.05e5·23-s − 1.54e6·25-s − 5.44e5·26-s + 1.37e6·28-s − 5.68e5·29-s − 6.46e6·31-s + 5.82e6·32-s − 2.19e6·34-s + 2.09e6·35-s − 1.41e7·37-s − 2.48e6·38-s + 5.70e6·40-s + 1.35e7·41-s + ⋯ |
| L(s) = 1 | + 0.425·2-s − 0.818·4-s − 0.455·5-s − 0.517·7-s − 0.774·8-s − 0.193·10-s + 1.48·11-s − 0.548·13-s − 0.220·14-s + 0.489·16-s − 0.660·17-s − 0.454·19-s + 0.372·20-s + 0.630·22-s + 0.153·23-s − 0.792·25-s − 0.233·26-s + 0.423·28-s − 0.149·29-s − 1.25·31-s + 0.982·32-s − 0.281·34-s + 0.235·35-s − 1.23·37-s − 0.193·38-s + 0.352·40-s + 0.750·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8313513193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8313513193\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + 3.41e6T \) |
| good | 2 | \( 1 - 9.62T + 512T^{2} \) |
| 5 | \( 1 + 636.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.28e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.18e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.64e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.27e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.58e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.05e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.68e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.46e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.41e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.35e7T + 3.27e14T^{2} \) |
| 47 | \( 1 + 2.14e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.02e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.16e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.09e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.24e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.04e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.16e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.47e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.15e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.39e8T + 7.60e17T^{2} \) |
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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
−9.456599669636930645746187602744, −9.151597057501776458849011764511, −8.033094349807727660696253115770, −6.86334842943389738257482443654, −6.02558528399250651000538193944, −4.83357798125777602601820610551, −3.98575569677555560520861155426, −3.29856611495874161584844274227, −1.76237648562871771703299605498, −0.35501414351202598114663097261,
0.35501414351202598114663097261, 1.76237648562871771703299605498, 3.29856611495874161584844274227, 3.98575569677555560520861155426, 4.83357798125777602601820610551, 6.02558528399250651000538193944, 6.86334842943389738257482443654, 8.033094349807727660696253115770, 9.151597057501776458849011764511, 9.456599669636930645746187602744
