| L(s) = 1 | − 2.18·2-s + 3-s + 2.75·4-s − 3.64·5-s − 2.18·6-s + 2.94·7-s − 1.64·8-s + 9-s + 7.95·10-s − 2.90·11-s + 2.75·12-s + 1.83·13-s − 6.42·14-s − 3.64·15-s − 1.91·16-s + 6.16·17-s − 2.18·18-s − 7.26·19-s − 10.0·20-s + 2.94·21-s + 6.32·22-s − 23-s − 1.64·24-s + 8.30·25-s − 4.00·26-s + 27-s + 8.12·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 0.577·3-s + 1.37·4-s − 1.63·5-s − 0.890·6-s + 1.11·7-s − 0.582·8-s + 0.333·9-s + 2.51·10-s − 0.874·11-s + 0.795·12-s + 0.508·13-s − 1.71·14-s − 0.941·15-s − 0.479·16-s + 1.49·17-s − 0.514·18-s − 1.66·19-s − 2.24·20-s + 0.643·21-s + 1.34·22-s − 0.208·23-s − 0.336·24-s + 1.66·25-s − 0.784·26-s + 0.192·27-s + 1.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7196344074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7196344074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 + 7.26T + 19T^{2} \) |
| 31 | \( 1 - 9.78T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 + 6.05T + 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 + 6.40T + 47T^{2} \) |
| 53 | \( 1 - 4.62T + 53T^{2} \) |
| 59 | \( 1 + 3.54T + 59T^{2} \) |
| 61 | \( 1 + 9.97T + 61T^{2} \) |
| 67 | \( 1 - 5.17T + 67T^{2} \) |
| 71 | \( 1 + 4.85T + 71T^{2} \) |
| 73 | \( 1 - 7.56T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 3.11T + 89T^{2} \) |
| 97 | \( 1 + 1.36T + 97T^{2} \) |
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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
−8.752866132158578013211829739428, −8.271972802198237875031214638813, −7.925021403215995179632467079620, −7.46914363531034700310604712878, −6.41448644065048258610657277444, −4.91559931772723679991289092079, −4.17076336121403788041424756049, −3.08745180053354099013511869588, −1.89261608490204527452089851616, −0.70127434368208780426885498877,
0.70127434368208780426885498877, 1.89261608490204527452089851616, 3.08745180053354099013511869588, 4.17076336121403788041424756049, 4.91559931772723679991289092079, 6.41448644065048258610657277444, 7.46914363531034700310604712878, 7.925021403215995179632467079620, 8.271972802198237875031214638813, 8.752866132158578013211829739428
