| L(s) = 1 | − 1.31·2-s − 0.266·4-s + 4.78·7-s + 2.98·8-s − 0.621·11-s + 5.28·13-s − 6.30·14-s − 3.39·16-s + 1.89·17-s + 5.01·19-s + 0.818·22-s + 7.01·23-s − 6.95·26-s − 1.27·28-s + 2.99·29-s + 31-s − 1.49·32-s − 2.49·34-s − 4.38·37-s − 6.60·38-s − 6.36·41-s + 12.0·43-s + 0.165·44-s − 9.23·46-s + 5.34·47-s + 15.9·49-s − 1.40·52-s + ⋯ |
| L(s) = 1 | − 0.930·2-s − 0.133·4-s + 1.81·7-s + 1.05·8-s − 0.187·11-s + 1.46·13-s − 1.68·14-s − 0.848·16-s + 0.458·17-s + 1.15·19-s + 0.174·22-s + 1.46·23-s − 1.36·26-s − 0.241·28-s + 0.555·29-s + 0.179·31-s − 0.264·32-s − 0.427·34-s − 0.720·37-s − 1.07·38-s − 0.994·41-s + 1.83·43-s + 0.0250·44-s − 1.36·46-s + 0.779·47-s + 2.27·49-s − 0.195·52-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
| \(L(1)\) |
\(\approx\) |
\(1.889840951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.889840951\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.31T + 2T^{2} \) |
| 7 | \( 1 - 4.78T + 7T^{2} \) |
| 11 | \( 1 + 0.621T + 11T^{2} \) |
| 13 | \( 1 - 5.28T + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 - 5.01T + 19T^{2} \) |
| 23 | \( 1 - 7.01T + 23T^{2} \) |
| 29 | \( 1 - 2.99T + 29T^{2} \) |
| 37 | \( 1 + 4.38T + 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 0.500T + 61T^{2} \) |
| 67 | \( 1 - 2.30T + 67T^{2} \) |
| 71 | \( 1 + 5.35T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.255771460902473198452584327107, −7.41081828531312158996826899982, −6.95317662588045493577704655111, −5.57422865619186386267379038323, −5.24595160008819807370788926525, −4.40241906305682852978346418438, −3.67397560221138510764590432908, −2.46735942659544697980545355242, −1.22598106821792188454460755690, −1.08424006240325558730626633936,
1.08424006240325558730626633936, 1.22598106821792188454460755690, 2.46735942659544697980545355242, 3.67397560221138510764590432908, 4.40241906305682852978346418438, 5.24595160008819807370788926525, 5.57422865619186386267379038323, 6.95317662588045493577704655111, 7.41081828531312158996826899982, 8.255771460902473198452584327107
