| L(s) = 1 | + 652.·2-s + 4.14e3·3-s + 2.95e5·4-s + 3.58e5·5-s + 2.70e6·6-s + 1.05e7·7-s + 1.07e8·8-s − 1.11e8·9-s + 2.33e8·10-s + 1.15e9·11-s + 1.22e9·12-s − 2.51e9·13-s + 6.86e9·14-s + 1.48e9·15-s + 3.12e10·16-s + 1.45e10·17-s − 7.31e10·18-s + 1.40e11·19-s + 1.05e11·20-s + 4.35e10·21-s + 7.55e11·22-s − 5.22e11·23-s + 4.43e11·24-s − 6.34e11·25-s − 1.64e12·26-s − 9.98e11·27-s + 3.10e12·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 0.364·3-s + 2.25·4-s + 0.410·5-s + 0.657·6-s + 0.689·7-s + 2.25·8-s − 0.867·9-s + 0.739·10-s + 1.62·11-s + 0.820·12-s − 0.855·13-s + 1.24·14-s + 0.149·15-s + 1.81·16-s + 0.505·17-s − 1.56·18-s + 1.89·19-s + 0.923·20-s + 0.251·21-s + 2.93·22-s − 1.38·23-s + 0.822·24-s − 0.831·25-s − 1.54·26-s − 0.680·27-s + 1.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(8.829044625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.829044625\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 5.00e11T \) |
| good | 2 | \( 1 - 652.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 4.14e3T + 1.29e8T^{2} \) |
| 5 | \( 1 - 3.58e5T + 7.62e11T^{2} \) |
| 7 | \( 1 - 1.05e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.15e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.51e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.45e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.40e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 5.22e11T + 1.41e23T^{2} \) |
| 31 | \( 1 - 5.44e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.38e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 5.39e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 5.68e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.71e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 6.93e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.52e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.52e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 9.07e14T + 1.10e31T^{2} \) |
| 71 | \( 1 - 1.43e14T + 2.96e31T^{2} \) |
| 73 | \( 1 - 5.43e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 6.13e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.62e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 4.92e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 7.93e16T + 5.95e33T^{2} \) |
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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
−13.83951308613720945278213511974, −11.93289676419376207477366288457, −11.67938219536357048020223748134, −9.666644977983306039066054646862, −7.77678020980747930210900299743, −6.27605545273809030954142438361, −5.25066688433097727535377404318, −3.96123332308021918544349544055, −2.77672449496463176858089199123, −1.52266995033413137981194983562,
1.52266995033413137981194983562, 2.77672449496463176858089199123, 3.96123332308021918544349544055, 5.25066688433097727535377404318, 6.27605545273809030954142438361, 7.77678020980747930210900299743, 9.666644977983306039066054646862, 11.67938219536357048020223748134, 11.93289676419376207477366288457, 13.83951308613720945278213511974
