| L(s) = 1 | + 687.·2-s − 1.67e4·3-s + 3.42e5·4-s + 3.76e5·5-s − 1.15e7·6-s − 3.08e5·7-s + 1.45e8·8-s + 1.50e8·9-s + 2.58e8·10-s − 4.41e8·11-s − 5.72e9·12-s + 7.99e8·13-s − 2.12e8·14-s − 6.29e9·15-s + 5.50e10·16-s + 3.92e10·17-s + 1.03e11·18-s − 3.46e10·19-s + 1.28e11·20-s + 5.16e9·21-s − 3.04e11·22-s + 5.56e11·23-s − 2.42e12·24-s − 6.21e11·25-s + 5.49e11·26-s − 3.61e11·27-s − 1.05e11·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 1.47·3-s + 2.61·4-s + 0.430·5-s − 2.79·6-s − 0.0202·7-s + 3.05·8-s + 1.16·9-s + 0.818·10-s − 0.621·11-s − 3.84·12-s + 0.271·13-s − 0.0384·14-s − 0.634·15-s + 3.20·16-s + 1.36·17-s + 2.21·18-s − 0.468·19-s + 1.12·20-s + 0.0297·21-s − 1.18·22-s + 1.48·23-s − 4.50·24-s − 0.814·25-s + 0.516·26-s − 0.246·27-s − 0.0528·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(5.180994552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.180994552\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 5.00e11T \) |
| good | 2 | \( 1 - 687.T + 1.31e5T^{2} \) |
| 3 | \( 1 + 1.67e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 3.76e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 3.08e5T + 2.32e14T^{2} \) |
| 11 | \( 1 + 4.41e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 7.99e8T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.92e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 3.46e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 5.56e11T + 1.41e23T^{2} \) |
| 31 | \( 1 - 3.74e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.26e12T + 4.56e26T^{2} \) |
| 41 | \( 1 - 9.56e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 8.89e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.56e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.50e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.81e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.40e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 9.70e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 1.90e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.33e16T + 4.74e31T^{2} \) |
| 79 | \( 1 + 7.46e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.07e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 4.42e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 4.33e16T + 5.95e33T^{2} \) |
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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
−13.07281412439523159356090698840, −12.28130908365082609171405248920, −11.24982818895525411655431785468, −10.35829300469405874089125353319, −7.31023719075133093606078918019, −6.01403445059920282567193788097, −5.48300085007495679280062430594, −4.35011910243013021312621994522, −2.75672619280495870086511367300, −1.09552311392670621594912381226,
1.09552311392670621594912381226, 2.75672619280495870086511367300, 4.35011910243013021312621994522, 5.48300085007495679280062430594, 6.01403445059920282567193788097, 7.31023719075133093606078918019, 10.35829300469405874089125353319, 11.24982818895525411655431785468, 12.28130908365082609171405248920, 13.07281412439523159356090698840
