| L(s) = 1 | + 512·2-s + 1.08e3·3-s + 2.62e5·4-s − 5.97e6·5-s + 5.57e5·6-s + 1.67e8·7-s + 1.34e8·8-s − 1.16e9·9-s − 3.05e9·10-s + 1.40e10·11-s + 2.85e8·12-s + 6.63e10·13-s + 8.57e10·14-s − 6.51e9·15-s + 6.87e10·16-s − 2.32e10·17-s − 5.94e11·18-s − 1.03e12·19-s − 1.56e12·20-s + 1.82e11·21-s + 7.21e12·22-s − 1.07e13·23-s + 1.46e11·24-s + 1.66e13·25-s + 3.39e13·26-s − 2.53e12·27-s + 4.39e13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0319·3-s + 0.5·4-s − 1.36·5-s + 0.0226·6-s + 1.56·7-s + 0.353·8-s − 0.998·9-s − 0.967·10-s + 1.80·11-s + 0.0159·12-s + 1.73·13-s + 1.10·14-s − 0.0437·15-s + 0.250·16-s − 0.0476·17-s − 0.706·18-s − 0.733·19-s − 0.683·20-s + 0.0501·21-s + 1.27·22-s − 1.24·23-s + 0.0113·24-s + 0.871·25-s + 1.22·26-s − 0.0638·27-s + 0.784·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(3.844003918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.844003918\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 512T \) |
| 29 | \( 1 - 1.45e13T \) |
| good | 3 | \( 1 - 1.08e3T + 1.16e9T^{2} \) |
| 5 | \( 1 + 5.97e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.67e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.40e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 6.63e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 2.32e10T + 2.39e23T^{2} \) |
| 19 | \( 1 + 1.03e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.07e13T + 7.46e25T^{2} \) |
| 31 | \( 1 + 1.41e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.81e13T + 6.24e29T^{2} \) |
| 41 | \( 1 + 1.58e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 2.46e15T + 1.08e31T^{2} \) |
| 47 | \( 1 - 1.31e16T + 5.88e31T^{2} \) |
| 53 | \( 1 - 3.23e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 6.06e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.05e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 3.34e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 6.40e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 4.40e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.88e18T + 1.13e36T^{2} \) |
| 83 | \( 1 - 1.31e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 1.72e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 4.35e18T + 5.60e37T^{2} \) |
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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
−11.47887011211174347145807257166, −10.99989873734802126587138796101, −8.630615339482894312469793029412, −8.195232356796423444011280521182, −6.74728009860828782347925901552, −5.54076905515099768373753821774, −4.00121669973586658255756746655, −3.82970984383217545194459844243, −1.93977232461628177690931177536, −0.840922233275492410536014279690,
0.840922233275492410536014279690, 1.93977232461628177690931177536, 3.82970984383217545194459844243, 4.00121669973586658255756746655, 5.54076905515099768373753821774, 6.74728009860828782347925901552, 8.195232356796423444011280521182, 8.630615339482894312469793029412, 10.99989873734802126587138796101, 11.47887011211174347145807257166
