| L(s) = 1 | + 512·2-s + 5.10e4·3-s + 2.62e5·4-s + 1.95e6·5-s + 2.61e7·6-s + 2.72e7·7-s + 1.34e8·8-s + 1.44e9·9-s + 1.00e9·10-s + 8.90e9·11-s + 1.33e10·12-s − 3.88e10·13-s + 1.39e10·14-s + 9.97e10·15-s + 6.87e10·16-s − 4.92e11·17-s + 7.40e11·18-s + 7.02e11·19-s + 5.12e11·20-s + 1.39e12·21-s + 4.56e12·22-s − 5.93e12·23-s + 6.85e12·24-s + 3.81e12·25-s − 1.99e13·26-s + 1.44e13·27-s + 7.14e12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.49·3-s + 0.5·4-s + 0.447·5-s + 1.05·6-s + 0.255·7-s + 0.353·8-s + 1.24·9-s + 0.316·10-s + 1.13·11-s + 0.749·12-s − 1.01·13-s + 0.180·14-s + 0.669·15-s + 0.250·16-s − 1.00·17-s + 0.879·18-s + 0.499·19-s + 0.223·20-s + 0.382·21-s + 0.805·22-s − 0.686·23-s + 0.529·24-s + 0.199·25-s − 0.719·26-s + 0.365·27-s + 0.127·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(5.394346450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.394346450\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 512T \) |
| 5 | \( 1 - 1.95e6T \) |
| good | 3 | \( 1 - 5.10e4T + 1.16e9T^{2} \) |
| 7 | \( 1 - 2.72e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 8.90e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 3.88e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 4.92e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 7.02e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 5.93e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 1.42e14T + 6.10e27T^{2} \) |
| 31 | \( 1 - 2.10e14T + 2.16e28T^{2} \) |
| 37 | \( 1 + 1.00e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 4.08e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 3.21e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 6.44e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 1.88e15T + 5.77e32T^{2} \) |
| 59 | \( 1 + 4.28e15T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.19e17T + 8.34e33T^{2} \) |
| 67 | \( 1 - 1.93e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 3.68e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 9.09e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 5.25e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 2.77e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 3.88e17T + 1.09e37T^{2} \) |
| 97 | \( 1 + 9.73e17T + 5.60e37T^{2} \) |
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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
−15.53385468812915918999060100400, −14.33612100989044385513149005056, −13.70588276651564951966614978321, −12.03541337679875878004693680349, −9.860578206753873640962656382450, −8.444043481239405089174898302199, −6.75852607962359498526493161754, −4.53548926144809339788761660784, −2.98943631278692431912506676383, −1.75584299494003521194763748316,
1.75584299494003521194763748316, 2.98943631278692431912506676383, 4.53548926144809339788761660784, 6.75852607962359498526493161754, 8.444043481239405089174898302199, 9.860578206753873640962656382450, 12.03541337679875878004693680349, 13.70588276651564951966614978321, 14.33612100989044385513149005056, 15.53385468812915918999060100400
