| L(s) = 1 | + 3-s + 0.336·5-s + 0.568·7-s + 9-s + 5.54·11-s − 5.92·13-s + 0.336·15-s + 0.0480·17-s − 3.03·19-s + 0.568·21-s − 5.32·23-s − 4.88·25-s + 27-s + 0.918·29-s − 8.65·31-s + 5.54·33-s + 0.191·35-s − 6.04·37-s − 5.92·39-s + 1.72·41-s − 6.69·43-s + 0.336·45-s − 0.136·47-s − 6.67·49-s + 0.0480·51-s − 1.99·53-s + 1.86·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.150·5-s + 0.214·7-s + 0.333·9-s + 1.67·11-s − 1.64·13-s + 0.0868·15-s + 0.0116·17-s − 0.695·19-s + 0.124·21-s − 1.11·23-s − 0.977·25-s + 0.192·27-s + 0.170·29-s − 1.55·31-s + 0.964·33-s + 0.0323·35-s − 0.994·37-s − 0.949·39-s + 0.269·41-s − 1.02·43-s + 0.0501·45-s − 0.0199·47-s − 0.953·49-s + 0.00673·51-s − 0.274·53-s + 0.251·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 - 0.336T + 5T^{2} \) |
| 7 | \( 1 - 0.568T + 7T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 13 | \( 1 + 5.92T + 13T^{2} \) |
| 17 | \( 1 - 0.0480T + 17T^{2} \) |
| 19 | \( 1 + 3.03T + 19T^{2} \) |
| 23 | \( 1 + 5.32T + 23T^{2} \) |
| 29 | \( 1 - 0.918T + 29T^{2} \) |
| 31 | \( 1 + 8.65T + 31T^{2} \) |
| 37 | \( 1 + 6.04T + 37T^{2} \) |
| 41 | \( 1 - 1.72T + 41T^{2} \) |
| 43 | \( 1 + 6.69T + 43T^{2} \) |
| 47 | \( 1 + 0.136T + 47T^{2} \) |
| 53 | \( 1 + 1.99T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 6.84T + 61T^{2} \) |
| 67 | \( 1 - 7.26T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 5.77T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 5.52T + 83T^{2} \) |
| 89 | \( 1 + 7.27T + 89T^{2} \) |
| 97 | \( 1 - 6.99T + 97T^{2} \) |
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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
−7.70796078262627282142326324936, −7.06329266475039528642798969779, −6.43932300936957773160564618894, −5.58792232085051383810809940292, −4.67247917598012338099795607594, −4.01898199272857782304571283321, −3.30487011334769625253441705395, −2.11401428398453786493827221554, −1.66971185223449705088234789797, 0,
1.66971185223449705088234789797, 2.11401428398453786493827221554, 3.30487011334769625253441705395, 4.01898199272857782304571283321, 4.67247917598012338099795607594, 5.58792232085051383810809940292, 6.43932300936957773160564618894, 7.06329266475039528642798969779, 7.70796078262627282142326324936
