| L(s) = 1 | + 256·2-s + 2.18e4·3-s + 6.55e4·4-s − 3.90e5·5-s + 5.58e6·6-s + 1.53e7·7-s + 1.67e7·8-s + 3.47e8·9-s − 1.00e8·10-s − 1.15e9·11-s + 1.43e9·12-s + 4.71e8·13-s + 3.93e9·14-s − 8.52e9·15-s + 4.29e9·16-s + 1.89e10·17-s + 8.89e10·18-s − 7.11e10·19-s − 2.56e10·20-s + 3.35e11·21-s − 2.94e11·22-s − 6.03e10·23-s + 3.66e11·24-s + 1.52e11·25-s + 1.20e11·26-s + 4.76e12·27-s + 1.00e12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.92·3-s + 0.5·4-s − 0.447·5-s + 1.35·6-s + 1.00·7-s + 0.353·8-s + 2.68·9-s − 0.316·10-s − 1.61·11-s + 0.960·12-s + 0.160·13-s + 0.712·14-s − 0.859·15-s + 0.250·16-s + 0.658·17-s + 1.90·18-s − 0.960·19-s − 0.223·20-s + 1.93·21-s − 1.14·22-s − 0.160·23-s + 0.679·24-s + 0.200·25-s + 0.113·26-s + 3.24·27-s + 0.503·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(5.276514392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.276514392\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 256T \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 3 | \( 1 - 2.18e4T + 1.29e8T^{2} \) |
| 7 | \( 1 - 1.53e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.15e9T + 5.05e17T^{2} \) |
| 13 | \( 1 - 4.71e8T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.89e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 7.11e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 6.03e10T + 1.41e23T^{2} \) |
| 29 | \( 1 - 1.95e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 2.11e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.79e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 7.71e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 9.99e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 2.88e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 1.47e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 3.95e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.00e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 3.33e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 2.99e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.02e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 4.95e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.56e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.06e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 3.74e16T + 5.95e33T^{2} \) |
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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
−15.79977185819673406160331055604, −14.89257222253528925539165822730, −13.87900508727161083944734268257, −12.66806956792330355632198207896, −10.48194131361407213808657194346, −8.414702524300610678224362229570, −7.56453926402556833448452837594, −4.66794664716345343582458429224, −3.17904426674912031285594886645, −1.90031851112469128941490957302,
1.90031851112469128941490957302, 3.17904426674912031285594886645, 4.66794664716345343582458429224, 7.56453926402556833448452837594, 8.414702524300610678224362229570, 10.48194131361407213808657194346, 12.66806956792330355632198207896, 13.87900508727161083944734268257, 14.89257222253528925539165822730, 15.79977185819673406160331055604
