| L(s) = 1 | − 0.978·2-s − 1.04·4-s − 1.38·7-s + 2.97·8-s − 2.53·11-s − 4.49·13-s + 1.35·14-s − 0.830·16-s + 4.15·17-s − 5.63·19-s + 2.48·22-s − 7.13·23-s + 4.40·26-s + 1.44·28-s + 5.51·29-s − 31-s − 5.14·32-s − 4.07·34-s − 5.88·37-s + 5.51·38-s − 3.95·41-s + 7.65·43-s + 2.64·44-s + 6.98·46-s + 3.58·47-s − 5.08·49-s + 4.68·52-s + ⋯ |
| L(s) = 1 | − 0.692·2-s − 0.520·4-s − 0.523·7-s + 1.05·8-s − 0.764·11-s − 1.24·13-s + 0.362·14-s − 0.207·16-s + 1.00·17-s − 1.29·19-s + 0.529·22-s − 1.48·23-s + 0.863·26-s + 0.272·28-s + 1.02·29-s − 0.179·31-s − 0.908·32-s − 0.698·34-s − 0.966·37-s + 0.895·38-s − 0.617·41-s + 1.16·43-s + 0.398·44-s + 1.02·46-s + 0.522·47-s − 0.726·49-s + 0.649·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3590837113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3590837113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.978T + 2T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 - 4.15T + 17T^{2} \) |
| 19 | \( 1 + 5.63T + 19T^{2} \) |
| 23 | \( 1 + 7.13T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 + 3.95T + 41T^{2} \) |
| 43 | \( 1 - 7.65T + 43T^{2} \) |
| 47 | \( 1 - 3.58T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 4.53T + 71T^{2} \) |
| 73 | \( 1 - 4.68T + 73T^{2} \) |
| 79 | \( 1 - 0.0707T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−8.002417343613733256368350661519, −7.52765936796785577388945132330, −6.66551289893197581977008803818, −5.86545667648382090750231559541, −5.03433970140155794412610027957, −4.45997204081936308018079755347, −3.56949723655014657276888145688, −2.60425619917673845213243539643, −1.70727782313655806129837322304, −0.33285784568041947251405962414,
0.33285784568041947251405962414, 1.70727782313655806129837322304, 2.60425619917673845213243539643, 3.56949723655014657276888145688, 4.45997204081936308018079755347, 5.03433970140155794412610027957, 5.86545667648382090750231559541, 6.66551289893197581977008803818, 7.52765936796785577388945132330, 8.002417343613733256368350661519
