| L(s) = 1 | + 32·2-s − 141.·3-s + 1.02e3·4-s + 3.12e3·5-s − 4.53e3·6-s + 8.55e4·7-s + 3.27e4·8-s − 1.57e5·9-s + 1.00e5·10-s + 7.67e5·11-s − 1.45e5·12-s + 2.20e5·13-s + 2.73e6·14-s − 4.42e5·15-s + 1.04e6·16-s − 9.30e5·17-s − 5.02e6·18-s − 1.77e7·19-s + 3.20e6·20-s − 1.21e7·21-s + 2.45e7·22-s − 3.99e7·23-s − 4.64e6·24-s + 9.76e6·25-s + 7.07e6·26-s + 4.73e7·27-s + 8.76e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.336·3-s + 0.5·4-s + 0.447·5-s − 0.238·6-s + 1.92·7-s + 0.353·8-s − 0.886·9-s + 0.316·10-s + 1.43·11-s − 0.168·12-s + 0.165·13-s + 1.36·14-s − 0.150·15-s + 0.250·16-s − 0.158·17-s − 0.626·18-s − 1.64·19-s + 0.223·20-s − 0.648·21-s + 1.01·22-s − 1.29·23-s − 0.119·24-s + 0.199·25-s + 0.116·26-s + 0.635·27-s + 0.962·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.711734219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.711734219\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 5 | \( 1 - 3.12e3T \) |
| good | 3 | \( 1 + 141.T + 1.77e5T^{2} \) |
| 7 | \( 1 - 8.55e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 7.67e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.20e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 9.30e5T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.77e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.99e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 7.68e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.96e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.40e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.26e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.88e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.57e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.09e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.77e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.64e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.63e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.03e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 4.27e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.96e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.35e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.25e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.08e11T + 7.15e21T^{2} \) |
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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
−17.77736466117883358148823341509, −16.93640620326652268169203828394, −14.75357021979853328790989824818, −14.07432227422831341256009847652, −11.97272180346616251378113402968, −10.97230457851962835483947881259, −8.436532141699225614016171337797, −6.15015776511802460800897197579, −4.50776396697386038857954667873, −1.77322034821727295348141717469,
1.77322034821727295348141717469, 4.50776396697386038857954667873, 6.15015776511802460800897197579, 8.436532141699225614016171337797, 10.97230457851962835483947881259, 11.97272180346616251378113402968, 14.07432227422831341256009847652, 14.75357021979853328790989824818, 16.93640620326652268169203828394, 17.77736466117883358148823341509
