| L(s) = 1 | − 3.27e4·2-s + 1.82e7·3-s + 1.07e9·4-s − 3.05e10·5-s − 5.98e11·6-s − 9.80e12·7-s − 3.51e13·8-s − 2.83e14·9-s + 1.00e15·10-s − 2.50e16·11-s + 1.96e16·12-s − 5.10e16·13-s + 3.21e17·14-s − 5.57e17·15-s + 1.15e18·16-s + 5.39e17·17-s + 9.29e18·18-s + 8.95e19·19-s − 3.27e19·20-s − 1.79e20·21-s + 8.20e20·22-s + 3.91e20·23-s − 6.42e20·24-s + 9.31e20·25-s + 1.67e21·26-s − 1.64e22·27-s − 1.05e22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.735·3-s + 0.5·4-s − 0.447·5-s − 0.519·6-s − 0.780·7-s − 0.353·8-s − 0.459·9-s + 0.316·10-s − 1.80·11-s + 0.367·12-s − 0.276·13-s + 0.551·14-s − 0.328·15-s + 0.250·16-s + 0.0457·17-s + 0.324·18-s + 1.35·19-s − 0.223·20-s − 0.573·21-s + 1.27·22-s + 0.306·23-s − 0.259·24-s + 0.200·25-s + 0.195·26-s − 1.07·27-s − 0.390·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+31/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(16)\) |
\(\approx\) |
\(0.9434338758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9434338758\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3.27e4T \) |
| 5 | \( 1 + 3.05e10T \) |
| good | 3 | \( 1 - 1.82e7T + 6.17e14T^{2} \) |
| 7 | \( 1 + 9.80e12T + 1.57e26T^{2} \) |
| 11 | \( 1 + 2.50e16T + 1.91e32T^{2} \) |
| 13 | \( 1 + 5.10e16T + 3.40e34T^{2} \) |
| 17 | \( 1 - 5.39e17T + 1.39e38T^{2} \) |
| 19 | \( 1 - 8.95e19T + 4.37e39T^{2} \) |
| 23 | \( 1 - 3.91e20T + 1.63e42T^{2} \) |
| 29 | \( 1 - 7.65e22T + 2.15e45T^{2} \) |
| 31 | \( 1 + 1.67e23T + 1.70e46T^{2} \) |
| 37 | \( 1 + 2.35e24T + 4.11e48T^{2} \) |
| 41 | \( 1 - 4.23e24T + 9.91e49T^{2} \) |
| 43 | \( 1 - 2.21e25T + 4.34e50T^{2} \) |
| 47 | \( 1 - 9.02e25T + 6.83e51T^{2} \) |
| 53 | \( 1 - 8.66e25T + 2.83e53T^{2} \) |
| 59 | \( 1 - 3.37e27T + 7.87e54T^{2} \) |
| 61 | \( 1 + 3.72e27T + 2.21e55T^{2} \) |
| 67 | \( 1 + 1.08e28T + 4.05e56T^{2} \) |
| 71 | \( 1 - 3.01e28T + 2.44e57T^{2} \) |
| 73 | \( 1 - 1.41e29T + 5.79e57T^{2} \) |
| 79 | \( 1 - 4.24e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 2.61e29T + 3.10e59T^{2} \) |
| 89 | \( 1 + 1.92e30T + 2.69e60T^{2} \) |
| 97 | \( 1 + 1.19e30T + 3.88e61T^{2} \) |
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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
−13.84203614138844073083266180466, −12.41105371858153765749923078374, −10.77638553179234885648872178839, −9.508695960247308912957377546135, −8.249202534134590961034378240617, −7.26941571540351477127699767725, −5.41981212374271320712452928847, −3.27712225625199937327068013358, −2.48150467550552921887619836060, −0.52924740275238647642561331536,
0.52924740275238647642561331536, 2.48150467550552921887619836060, 3.27712225625199937327068013358, 5.41981212374271320712452928847, 7.26941571540351477127699767725, 8.249202534134590961034378240617, 9.508695960247308912957377546135, 10.77638553179234885648872178839, 12.41105371858153765749923078374, 13.84203614138844073083266180466
