| L(s) = 1 | − 361.·2-s + 9.47e3·3-s − 465.·4-s + 2.15e5·5-s − 3.42e6·6-s − 1.26e7·7-s + 4.75e7·8-s − 3.92e7·9-s − 7.78e7·10-s − 2.56e8·11-s − 4.41e6·12-s − 2.85e9·13-s + 4.55e9·14-s + 2.04e9·15-s − 1.71e10·16-s + 4.22e10·17-s + 1.42e10·18-s + 8.56e10·19-s − 1.00e8·20-s − 1.19e11·21-s + 9.27e10·22-s − 3.49e11·23-s + 4.50e11·24-s − 7.16e11·25-s + 1.03e12·26-s − 1.59e12·27-s + 5.86e9·28-s + ⋯ |
| L(s) = 1 | − 0.998·2-s + 0.834·3-s − 0.00355·4-s + 0.246·5-s − 0.832·6-s − 0.827·7-s + 1.00·8-s − 0.304·9-s − 0.246·10-s − 0.361·11-s − 0.00296·12-s − 0.972·13-s + 0.825·14-s + 0.205·15-s − 0.996·16-s + 1.47·17-s + 0.303·18-s + 1.15·19-s − 0.000876·20-s − 0.689·21-s + 0.360·22-s − 0.930·23-s + 0.835·24-s − 0.939·25-s + 0.970·26-s − 1.08·27-s + 0.00293·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.053306244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.053306244\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 5.00e11T \) |
| good | 2 | \( 1 + 361.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 9.47e3T + 1.29e8T^{2} \) |
| 5 | \( 1 - 2.15e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.26e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 2.56e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.85e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 4.22e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 8.56e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 3.49e11T + 1.41e23T^{2} \) |
| 31 | \( 1 - 3.88e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.35e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 8.27e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.16e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.09e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 5.38e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.96e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.36e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 9.41e14T + 1.10e31T^{2} \) |
| 71 | \( 1 - 1.53e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.27e16T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.05e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.73e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 1.48e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 8.75e16T + 5.95e33T^{2} \) |
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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
−13.54695398847437164383456421052, −12.03648652329841052587685321156, −10.04406003950884720525467468143, −9.624765629736311889759049288698, −8.305077380031588451264625969599, −7.38317645257475681440900559137, −5.45666180663762291262786192947, −3.53555109682943781684203912206, −2.23884164587679621990464540915, −0.62199670540909662391975821983,
0.62199670540909662391975821983, 2.23884164587679621990464540915, 3.53555109682943781684203912206, 5.45666180663762291262786192947, 7.38317645257475681440900559137, 8.305077380031588451264625969599, 9.624765629736311889759049288698, 10.04406003950884720525467468143, 12.03648652329841052587685321156, 13.54695398847437164383456421052
