| L(s) = 1 | − 1.46·2-s + 1.51·3-s + 0.138·4-s − 5-s − 2.20·6-s + 4.07·7-s + 2.72·8-s − 0.718·9-s + 1.46·10-s − 0.621·11-s + 0.208·12-s − 5.09·13-s − 5.95·14-s − 1.51·15-s − 4.25·16-s − 0.266·17-s + 1.05·18-s − 0.138·20-s + 6.14·21-s + 0.908·22-s − 5.84·23-s + 4.11·24-s + 25-s + 7.44·26-s − 5.61·27-s + 0.563·28-s + 4.07·29-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.872·3-s + 0.0691·4-s − 0.447·5-s − 0.901·6-s + 1.53·7-s + 0.962·8-s − 0.239·9-s + 0.462·10-s − 0.187·11-s + 0.0603·12-s − 1.41·13-s − 1.59·14-s − 0.389·15-s − 1.06·16-s − 0.0646·17-s + 0.247·18-s − 0.0309·20-s + 1.34·21-s + 0.193·22-s − 1.21·23-s + 0.839·24-s + 0.200·25-s + 1.46·26-s − 1.08·27-s + 0.106·28-s + 0.757·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 3 | \( 1 - 1.51T + 3T^{2} \) |
| 7 | \( 1 - 4.07T + 7T^{2} \) |
| 11 | \( 1 + 0.621T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 + 0.266T + 17T^{2} \) |
| 23 | \( 1 + 5.84T + 23T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 + 8.83T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 + 1.80T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 2.52T + 53T^{2} \) |
| 59 | \( 1 - 0.890T + 59T^{2} \) |
| 61 | \( 1 - 2.16T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 1.64T + 71T^{2} \) |
| 73 | \( 1 - 1.79T + 73T^{2} \) |
| 79 | \( 1 - 5.14T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 0.000747T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.753646768962158900227579059058, −8.149252234510423767032424831598, −7.71299117494297897427729698463, −7.08224793756223313770556726133, −5.42806065558138600207273869327, −4.72420657024288781718876544746, −3.83738665181085116790901100380, −2.45023617452399897959320134560, −1.66838672209491873278737338578, 0,
1.66838672209491873278737338578, 2.45023617452399897959320134560, 3.83738665181085116790901100380, 4.72420657024288781718876544746, 5.42806065558138600207273869327, 7.08224793756223313770556726133, 7.71299117494297897427729698463, 8.149252234510423767032424831598, 8.753646768962158900227579059058
