| L(s) = 1 | + 2.04e3·2-s − 3.39e5·3-s + 4.19e6·4-s − 4.88e7·5-s − 6.94e8·6-s + 5.67e9·7-s + 8.58e9·8-s + 2.08e10·9-s − 1.00e11·10-s − 7.29e10·11-s − 1.42e12·12-s + 7.50e12·13-s + 1.16e13·14-s + 1.65e13·15-s + 1.75e13·16-s − 1.22e14·17-s + 4.27e13·18-s − 6.14e14·19-s − 2.04e14·20-s − 1.92e15·21-s − 1.49e14·22-s − 3.69e15·23-s − 2.91e15·24-s + 2.38e15·25-s + 1.53e16·26-s + 2.48e16·27-s + 2.37e16·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.10·3-s + 0.5·4-s − 0.447·5-s − 0.781·6-s + 1.08·7-s + 0.353·8-s + 0.221·9-s − 0.316·10-s − 0.0770·11-s − 0.552·12-s + 1.16·13-s + 0.766·14-s + 0.494·15-s + 0.250·16-s − 0.868·17-s + 0.156·18-s − 1.20·19-s − 0.223·20-s − 1.19·21-s − 0.0544·22-s − 0.808·23-s − 0.390·24-s + 0.200·25-s + 0.821·26-s + 0.860·27-s + 0.542·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.04e3T \) |
| 5 | \( 1 + 4.88e7T \) |
| good | 3 | \( 1 + 3.39e5T + 9.41e10T^{2} \) |
| 7 | \( 1 - 5.67e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 7.29e10T + 8.95e23T^{2} \) |
| 13 | \( 1 - 7.50e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.22e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 6.14e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 3.69e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 1.00e17T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.10e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 5.06e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 5.37e17T + 1.24e37T^{2} \) |
| 43 | \( 1 + 3.30e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.83e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 4.33e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.82e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 5.31e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.46e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 3.40e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 1.21e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 3.86e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.91e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 1.38e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 3.39e21T + 4.96e45T^{2} \) |
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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
−14.55430448726141694530821031557, −12.92354923669322687883018516492, −11.43494994566668803452833107465, −10.94698648401794219119414465327, −8.288694230353171644476465211036, −6.49645296786283932791073601534, −5.23431630705502910321398833247, −3.98167045682760011718907147567, −1.75844881679893128551930352218, 0,
1.75844881679893128551930352218, 3.98167045682760011718907147567, 5.23431630705502910321398833247, 6.49645296786283932791073601534, 8.288694230353171644476465211036, 10.94698648401794219119414465327, 11.43494994566668803452833107465, 12.92354923669322687883018516492, 14.55430448726141694530821031557
