L(s) = 1 | + 9.69e3·2-s − 8.94e5·3-s + 6.04e7·4-s − 4.19e8·5-s − 8.67e9·6-s + 2.61e11·8-s − 4.78e10·9-s − 4.07e12·10-s + 1.33e12·11-s − 5.40e13·12-s − 6.92e13·13-s + 3.75e14·15-s + 5.02e14·16-s − 3.99e15·17-s − 4.63e14·18-s − 3.21e15·19-s − 2.53e16·20-s + 1.29e16·22-s + 6.09e16·23-s − 2.33e17·24-s − 1.21e17·25-s − 6.71e17·26-s + 8.00e17·27-s + 3.08e18·29-s + 3.64e18·30-s + 6.57e18·31-s − 3.88e18·32-s + ⋯ |
L(s) = 1 | + 1.67·2-s − 0.971·3-s + 1.80·4-s − 0.769·5-s − 1.62·6-s + 1.34·8-s − 0.0564·9-s − 1.28·10-s + 0.128·11-s − 1.75·12-s − 0.824·13-s + 0.747·15-s + 0.446·16-s − 1.66·17-s − 0.0944·18-s − 0.333·19-s − 1.38·20-s + 0.214·22-s + 0.580·23-s − 1.30·24-s − 0.408·25-s − 1.37·26-s + 1.02·27-s + 1.61·29-s + 1.25·30-s + 1.50·31-s − 0.596·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(13)\) |
\(\approx\) |
\(2.212057929\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.212057929\) |
\(L(\frac{27}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 - 9.69e3T + 3.35e7T^{2} \) |
| 3 | \( 1 + 8.94e5T + 8.47e11T^{2} \) |
| 5 | \( 1 + 4.19e8T + 2.98e17T^{2} \) |
| 11 | \( 1 - 1.33e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 6.92e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 3.99e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 3.21e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 6.09e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 3.08e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 6.57e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 3.76e18T + 1.60e39T^{2} \) |
| 41 | \( 1 + 2.38e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 2.96e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 8.07e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 4.02e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 6.81e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 1.82e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 1.02e23T + 4.48e45T^{2} \) |
| 71 | \( 1 + 6.11e22T + 1.91e46T^{2} \) |
| 73 | \( 1 - 2.19e21T + 3.82e46T^{2} \) |
| 79 | \( 1 + 1.84e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.18e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 2.36e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 5.26e24T + 4.66e49T^{2} \) |
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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
−11.63726754503515436737902391483, −10.39765315567138527675534856094, −8.467634837924676794400847585553, −6.86155293528280022642402563208, −6.31414783520212630008759120469, −4.92298945579584985241521823621, −4.57664832362640297178816426382, −3.27987036343785529466418275000, −2.22152589158804324214045879382, −0.48071225608662265303255322401,
0.48071225608662265303255322401, 2.22152589158804324214045879382, 3.27987036343785529466418275000, 4.57664832362640297178816426382, 4.92298945579584985241521823621, 6.31414783520212630008759120469, 6.86155293528280022642402563208, 8.467634837924676794400847585553, 10.39765315567138527675534856094, 11.63726754503515436737902391483