L(s) = 1 | + 5.84e6·2-s − 1.04e10·3-s + 2.53e13·4-s + 1.12e15·5-s − 6.10e16·6-s − 1.55e18·7-s + 9.64e19·8-s + 1.09e20·9-s + 6.55e21·10-s + 2.50e22·11-s − 2.64e23·12-s + 2.47e23·13-s − 9.06e24·14-s − 1.17e25·15-s + 3.40e26·16-s − 1.03e26·17-s + 6.39e26·18-s − 6.03e26·19-s + 2.84e28·20-s + 1.62e28·21-s + 1.46e29·22-s − 3.77e28·23-s − 1.00e30·24-s + 1.22e29·25-s + 1.44e30·26-s − 1.14e30·27-s − 3.92e31·28-s + ⋯ |
L(s) = 1 | + 1.96·2-s − 0.577·3-s + 2.87·4-s + 1.05·5-s − 1.13·6-s − 1.05·7-s + 3.69·8-s + 0.333·9-s + 2.07·10-s + 1.02·11-s − 1.66·12-s + 0.278·13-s − 2.06·14-s − 0.607·15-s + 4.40·16-s − 0.364·17-s + 0.656·18-s − 0.194·19-s + 3.02·20-s + 0.606·21-s + 2.00·22-s − 0.199·23-s − 2.13·24-s + 0.107·25-s + 0.548·26-s − 0.192·27-s − 3.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(22)\) |
\(\approx\) |
\(7.201617652\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.201617652\) |
\(L(\frac{45}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.04e10T \) |
good | 2 | \( 1 - 5.84e6T + 8.79e12T^{2} \) |
| 5 | \( 1 - 1.12e15T + 1.13e30T^{2} \) |
| 7 | \( 1 + 1.55e18T + 2.18e36T^{2} \) |
| 11 | \( 1 - 2.50e22T + 6.02e44T^{2} \) |
| 13 | \( 1 - 2.47e23T + 7.93e47T^{2} \) |
| 17 | \( 1 + 1.03e26T + 8.11e52T^{2} \) |
| 19 | \( 1 + 6.03e26T + 9.69e54T^{2} \) |
| 23 | \( 1 + 3.77e28T + 3.58e58T^{2} \) |
| 29 | \( 1 - 4.08e31T + 7.64e62T^{2} \) |
| 31 | \( 1 - 1.22e32T + 1.34e64T^{2} \) |
| 37 | \( 1 - 1.91e33T + 2.70e67T^{2} \) |
| 41 | \( 1 + 5.34e34T + 2.23e69T^{2} \) |
| 43 | \( 1 + 3.81e33T + 1.73e70T^{2} \) |
| 47 | \( 1 + 9.23e35T + 7.94e71T^{2} \) |
| 53 | \( 1 + 7.06e36T + 1.39e74T^{2} \) |
| 59 | \( 1 + 7.34e37T + 1.40e76T^{2} \) |
| 61 | \( 1 + 4.55e38T + 5.87e76T^{2} \) |
| 67 | \( 1 - 9.19e38T + 3.32e78T^{2} \) |
| 71 | \( 1 + 1.09e40T + 4.01e79T^{2} \) |
| 73 | \( 1 - 6.59e39T + 1.32e80T^{2} \) |
| 79 | \( 1 + 1.10e41T + 3.96e81T^{2} \) |
| 83 | \( 1 + 1.89e41T + 3.31e82T^{2} \) |
| 89 | \( 1 + 1.34e41T + 6.66e83T^{2} \) |
| 97 | \( 1 - 7.60e42T + 2.69e85T^{2} \) |
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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
−15.72480453218730449203495755705, −14.05297771016825676554077550617, −13.04046736872740678616415316059, −11.79990250537731085206369531976, −10.15521852414179608341851018729, −6.57600892713768839894895943071, −6.08189191622170466475631181701, −4.53928643921455225669044805885, −3.05741398354729471616315020819, −1.55444891737454959864496374162,
1.55444891737454959864496374162, 3.05741398354729471616315020819, 4.53928643921455225669044805885, 6.08189191622170466475631181701, 6.57600892713768839894895943071, 10.15521852414179608341851018729, 11.79990250537731085206369531976, 13.04046736872740678616415316059, 14.05297771016825676554077550617, 15.72480453218730449203495755705