| L(s) = 1 | − 256·2-s − 9.31e3·3-s + 6.55e4·4-s + 3.90e5·5-s + 2.38e6·6-s − 2.48e7·7-s − 1.67e7·8-s − 4.24e7·9-s − 1.00e8·10-s − 1.90e8·11-s − 6.10e8·12-s − 1.57e7·13-s + 6.36e9·14-s − 3.63e9·15-s + 4.29e9·16-s − 2.25e10·17-s + 1.08e10·18-s + 1.27e11·19-s + 2.56e10·20-s + 2.31e11·21-s + 4.87e10·22-s + 2.59e11·23-s + 1.56e11·24-s + 1.52e11·25-s + 4.02e9·26-s + 1.59e12·27-s − 1.62e12·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.819·3-s + 0.5·4-s + 0.447·5-s + 0.579·6-s − 1.63·7-s − 0.353·8-s − 0.328·9-s − 0.316·10-s − 0.267·11-s − 0.409·12-s − 0.00535·13-s + 1.15·14-s − 0.366·15-s + 0.250·16-s − 0.784·17-s + 0.232·18-s + 1.72·19-s + 0.223·20-s + 1.33·21-s + 0.189·22-s + 0.690·23-s + 0.289·24-s + 0.200·25-s + 0.00378·26-s + 1.08·27-s − 0.815·28-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
| \(L(9)\) |
\(\approx\) |
\(0.6384991665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6384991665\) |
| \(L(\frac{19}{2})\) |
|
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 | \( 1 + 9.31e3T + 1.29e8T^{2} \) |
| 7 | \( 1 + 2.48e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.90e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.57e7T + 8.65e18T^{2} \) |
| 17 | \( 1 + 2.25e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.27e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 2.59e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.57e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 7.91e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.09e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 1.98e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 4.60e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.66e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 4.00e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.82e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.48e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.71e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 3.87e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 2.85e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 2.80e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 7.04e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 5.51e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 6.42e16T + 5.95e33T^{2} \) |
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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
−16.71190426901195853340748588141, −15.70921941218575105644057520023, −13.43605969509634101210197496410, −11.90456567292496176224001549305, −10.37125833178106857491133091068, −9.156191399489473612719585899751, −6.89507437872306717036684652840, −5.66588688571795440059821513094, −2.91054587554933684561382951429, −0.62687264407191403541721810399,
0.62687264407191403541721810399, 2.91054587554933684561382951429, 5.66588688571795440059821513094, 6.89507437872306717036684652840, 9.156191399489473612719585899751, 10.37125833178106857491133091068, 11.90456567292496176224001549305, 13.43605969509634101210197496410, 15.70921941218575105644057520023, 16.71190426901195853340748588141
