| L(s) = 1 | − 0.638·2-s + 3-s − 1.59·4-s − 2.55·5-s − 0.638·6-s − 2.98·7-s + 2.29·8-s + 9-s + 1.63·10-s + 11-s − 1.59·12-s − 1.61·13-s + 1.90·14-s − 2.55·15-s + 1.71·16-s − 5.85·17-s − 0.638·18-s + 0.125·19-s + 4.06·20-s − 2.98·21-s − 0.638·22-s − 3.09·23-s + 2.29·24-s + 1.51·25-s + 1.03·26-s + 27-s + 4.74·28-s + ⋯ |
| L(s) = 1 | − 0.451·2-s + 0.577·3-s − 0.795·4-s − 1.14·5-s − 0.260·6-s − 1.12·7-s + 0.811·8-s + 0.333·9-s + 0.515·10-s + 0.301·11-s − 0.459·12-s − 0.447·13-s + 0.509·14-s − 0.658·15-s + 0.429·16-s − 1.42·17-s − 0.150·18-s + 0.0288·19-s + 0.908·20-s − 0.650·21-s − 0.136·22-s − 0.645·23-s + 0.468·24-s + 0.302·25-s + 0.202·26-s + 0.192·27-s + 0.897·28-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
| \(L(1)\) |
\(\approx\) |
\(0.5690461766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5690461766\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.638T + 2T^{2} \) |
| 5 | \( 1 + 2.55T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 + 5.85T + 17T^{2} \) |
| 19 | \( 1 - 0.125T + 19T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 - 0.191T + 29T^{2} \) |
| 31 | \( 1 + 1.81T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 - 6.37T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 + 0.116T + 47T^{2} \) |
| 53 | \( 1 - 8.43T + 53T^{2} \) |
| 59 | \( 1 + 0.197T + 59T^{2} \) |
| 67 | \( 1 - 2.32T + 67T^{2} \) |
| 71 | \( 1 + 4.03T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 2.21T + 79T^{2} \) |
| 83 | \( 1 - 9.40T + 83T^{2} \) |
| 89 | \( 1 - 4.97T + 89T^{2} \) |
| 97 | \( 1 + 0.00505T + 97T^{2} \) |
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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
−9.069620486738442191241268188773, −8.533027442333308556478844386122, −7.67450800369098680570576967792, −7.11329755784642818617858270322, −6.14882041223124052253249203863, −4.84052247946441641638422176177, −4.02615315654477802328012178309, −3.53349163659768816750983633304, −2.25728969849279894282134877999, −0.50470258475649847251120494107,
0.50470258475649847251120494107, 2.25728969849279894282134877999, 3.53349163659768816750983633304, 4.02615315654477802328012178309, 4.84052247946441641638422176177, 6.14882041223124052253249203863, 7.11329755784642818617858270322, 7.67450800369098680570576967792, 8.533027442333308556478844386122, 9.069620486738442191241268188773
