| L(s) = 1 | + 4.09e3·2-s + 6.18e5·3-s + 1.67e7·4-s − 2.44e8·5-s + 2.53e9·6-s + 4.78e10·7-s + 6.87e10·8-s − 4.64e11·9-s − 1.00e12·10-s + 3.58e11·11-s + 1.03e13·12-s + 7.76e13·13-s + 1.96e14·14-s − 1.50e14·15-s + 2.81e14·16-s + 1.50e15·17-s − 1.90e15·18-s + 4.83e15·19-s − 4.09e15·20-s + 2.96e16·21-s + 1.46e15·22-s + 2.59e16·23-s + 4.24e16·24-s + 5.96e16·25-s + 3.18e17·26-s − 8.11e17·27-s + 8.03e17·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.671·3-s + 0.5·4-s − 0.447·5-s + 0.475·6-s + 1.30·7-s + 0.353·8-s − 0.548·9-s − 0.316·10-s + 0.0344·11-s + 0.335·12-s + 0.924·13-s + 0.924·14-s − 0.300·15-s + 0.250·16-s + 0.627·17-s − 0.387·18-s + 0.501·19-s − 0.223·20-s + 0.878·21-s + 0.0243·22-s + 0.246·23-s + 0.237·24-s + 0.199·25-s + 0.653·26-s − 1.04·27-s + 0.653·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(4.518807412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.518807412\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.09e3T \) |
| 5 | \( 1 + 2.44e8T \) |
| good | 3 | \( 1 - 6.18e5T + 8.47e11T^{2} \) |
| 7 | \( 1 - 4.78e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 3.58e11T + 1.08e26T^{2} \) |
| 13 | \( 1 - 7.76e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 1.50e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 4.83e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 2.59e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 9.30e17T + 3.63e36T^{2} \) |
| 31 | \( 1 - 3.10e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 4.54e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.02e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 4.58e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 1.51e21T + 6.34e41T^{2} \) |
| 53 | \( 1 + 2.00e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 2.00e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.40e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 5.68e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 2.06e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + 3.20e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 4.28e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.11e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 1.75e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 1.28e25T + 4.66e49T^{2} \) |
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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
−14.56420477160205827152756985243, −13.79238779399364083481388158639, −12.00045798559841793275080789727, −10.90507687333009033194744043577, −8.674934187930878223447820595810, −7.59939937718354106091567344807, −5.61075416557292730241010202054, −4.13850937697760931376022353106, −2.79622925444236206197409075218, −1.23991409486712841662795472706,
1.23991409486712841662795472706, 2.79622925444236206197409075218, 4.13850937697760931376022353106, 5.61075416557292730241010202054, 7.59939937718354106091567344807, 8.674934187930878223447820595810, 10.90507687333009033194744043577, 12.00045798559841793275080789727, 13.79238779399364083481388158639, 14.56420477160205827152756985243
