| L(s) = 1 | − 256·2-s − 1.83e4·3-s + 6.55e4·4-s − 3.90e5·5-s + 4.69e6·6-s + 1.60e7·7-s − 1.67e7·8-s + 2.07e8·9-s + 1.00e8·10-s + 7.88e8·11-s − 1.20e9·12-s − 2.87e9·13-s − 4.11e9·14-s + 7.16e9·15-s + 4.29e9·16-s + 2.01e10·17-s − 5.31e10·18-s − 2.28e10·19-s − 2.56e10·20-s − 2.94e11·21-s − 2.01e11·22-s − 6.65e11·23-s + 3.07e11·24-s + 1.52e11·25-s + 7.35e11·26-s − 1.43e12·27-s + 1.05e12·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.61·3-s + 0.5·4-s − 0.447·5-s + 1.14·6-s + 1.05·7-s − 0.353·8-s + 1.60·9-s + 0.316·10-s + 1.10·11-s − 0.807·12-s − 0.977·13-s − 0.745·14-s + 0.721·15-s + 0.250·16-s + 0.700·17-s − 1.13·18-s − 0.308·19-s − 0.223·20-s − 1.70·21-s − 0.783·22-s − 1.77·23-s + 0.570·24-s + 0.200·25-s + 0.690·26-s − 0.978·27-s + 0.527·28-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
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
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 | \( 1 + 1.83e4T + 1.29e8T^{2} \) |
| 7 | \( 1 - 1.60e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 7.88e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.87e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 2.01e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 2.28e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 6.65e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.42e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 8.75e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.75e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 3.91e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.29e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + 3.00e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 1.01e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 9.67e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.14e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 4.04e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.35e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 6.99e14T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.08e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.23e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 2.21e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 9.41e16T + 5.95e33T^{2} \) |
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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
−16.44060109450586892843172728268, −14.75560750667871913677444486947, −12.03366484773966092165775879482, −11.49386042541845569500570278901, −10.00949088855829780957427183570, −7.83443484987907512307869010544, −6.24293214062582122949390919918, −4.58681730151072479810673840130, −1.41911298626837526464284591090, 0,
1.41911298626837526464284591090, 4.58681730151072479810673840130, 6.24293214062582122949390919918, 7.83443484987907512307869010544, 10.00949088855829780957427183570, 11.49386042541845569500570278901, 12.03366484773966092165775879482, 14.75560750667871913677444486947, 16.44060109450586892843172728268
