| L(s) = 1 | − 4.09e3·2-s − 7.01e5·3-s + 1.67e7·4-s − 2.44e8·5-s + 2.87e9·6-s − 7.10e8·7-s − 6.87e10·8-s − 3.54e11·9-s + 1.00e12·10-s + 6.69e12·11-s − 1.17e13·12-s − 4.06e12·13-s + 2.90e12·14-s + 1.71e14·15-s + 2.81e14·16-s + 2.05e15·17-s + 1.45e15·18-s + 1.32e16·19-s − 4.09e15·20-s + 4.98e14·21-s − 2.74e16·22-s − 2.83e16·23-s + 4.82e16·24-s + 5.96e16·25-s + 1.66e16·26-s + 8.43e17·27-s − 1.19e16·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.762·3-s + 0.5·4-s − 0.447·5-s + 0.539·6-s − 0.0193·7-s − 0.353·8-s − 0.418·9-s + 0.316·10-s + 0.642·11-s − 0.381·12-s − 0.0483·13-s + 0.0137·14-s + 0.341·15-s + 0.250·16-s + 0.855·17-s + 0.295·18-s + 1.37·19-s − 0.223·20-s + 0.0147·21-s − 0.454·22-s − 0.269·23-s + 0.269·24-s + 0.199·25-s + 0.0342·26-s + 1.08·27-s − 0.00969·28-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
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 + 7.01e5T + 8.47e11T^{2} \) |
| 7 | \( 1 + 7.10e8T + 1.34e21T^{2} \) |
| 11 | \( 1 - 6.69e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 4.06e12T + 7.05e27T^{2} \) |
| 17 | \( 1 - 2.05e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.32e16T + 9.30e31T^{2} \) |
| 23 | \( 1 + 2.83e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 8.79e17T + 3.63e36T^{2} \) |
| 31 | \( 1 + 3.20e17T + 1.92e37T^{2} \) |
| 37 | \( 1 + 4.64e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 1.18e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.34e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 7.78e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 3.18e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 2.03e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.16e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 4.98e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.16e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 1.07e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 2.51e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.47e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.69e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 4.56e24T + 4.66e49T^{2} \) |
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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
−14.30587599613203452273981982122, −12.14663262218137303556212539135, −11.32600843163379472079279165960, −9.832588671207865702563175028514, −8.257850168107581819790151845900, −6.76482799512350825123723644684, −5.30279706047964826348639516612, −3.28026065029470894403981133412, −1.25014930358597200731379863363, 0,
1.25014930358597200731379863363, 3.28026065029470894403981133412, 5.30279706047964826348639516612, 6.76482799512350825123723644684, 8.257850168107581819790151845900, 9.832588671207865702563175028514, 11.32600843163379472079279165960, 12.14663262218137303556212539135, 14.30587599613203452273981982122
