| L(s) = 1 | − 1.46·2-s − 3-s + 0.154·4-s + 3.84·5-s + 1.46·6-s + 2.46·7-s + 2.70·8-s + 9-s − 5.64·10-s + 11-s − 0.154·12-s − 4.64·13-s − 3.61·14-s − 3.84·15-s − 4.28·16-s + 5.70·17-s − 1.46·18-s + 6.71·19-s + 0.594·20-s − 2.46·21-s − 1.46·22-s + 2.93·23-s − 2.70·24-s + 9.76·25-s + 6.82·26-s − 27-s + 0.381·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.577·3-s + 0.0774·4-s + 1.71·5-s + 0.599·6-s + 0.930·7-s + 0.957·8-s + 0.333·9-s − 1.78·10-s + 0.301·11-s − 0.0447·12-s − 1.28·13-s − 0.965·14-s − 0.992·15-s − 1.07·16-s + 1.38·17-s − 0.345·18-s + 1.53·19-s + 0.133·20-s − 0.537·21-s − 0.312·22-s + 0.612·23-s − 0.552·24-s + 1.95·25-s + 1.33·26-s − 0.192·27-s + 0.0720·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.340524637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.340524637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 - 6.71T + 19T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 - 5.67T + 29T^{2} \) |
| 31 | \( 1 - 0.232T + 31T^{2} \) |
| 37 | \( 1 - 5.55T + 37T^{2} \) |
| 41 | \( 1 + 4.35T + 41T^{2} \) |
| 43 | \( 1 + 9.35T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 0.299T + 71T^{2} \) |
| 73 | \( 1 + 2.65T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 7.42T + 83T^{2} \) |
| 89 | \( 1 + 3.40T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−9.517195739909935267862231143846, −8.483581906920170190798486688171, −7.63159109757493681338399821717, −6.99805436942152077583131034807, −5.91932778202703455402622338795, −5.10248797750734925595742966860, −4.76122415113555541885687311618, −2.92446680688144743395273786845, −1.67533263074187084204463332169, −1.06048378224114871754722471483,
1.06048378224114871754722471483, 1.67533263074187084204463332169, 2.92446680688144743395273786845, 4.76122415113555541885687311618, 5.10248797750734925595742966860, 5.91932778202703455402622338795, 6.99805436942152077583131034807, 7.63159109757493681338399821717, 8.483581906920170190798486688171, 9.517195739909935267862231143846
