| L(s) = 1 | − 2-s + 2.29·3-s + 4-s + 2.69·5-s − 2.29·6-s − 8-s + 2.24·9-s − 2.69·10-s + 0.309·11-s + 2.29·12-s + 1.69·13-s + 6.18·15-s + 16-s + 4.15·17-s − 2.24·18-s − 6.09·19-s + 2.69·20-s − 0.309·22-s + 8.96·23-s − 2.29·24-s + 2.28·25-s − 1.69·26-s − 1.72·27-s − 2.55·29-s − 6.18·30-s − 4.49·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s + 1.20·5-s − 0.934·6-s − 0.353·8-s + 0.748·9-s − 0.853·10-s + 0.0932·11-s + 0.661·12-s + 0.469·13-s + 1.59·15-s + 0.250·16-s + 1.00·17-s − 0.529·18-s − 1.39·19-s + 0.603·20-s − 0.0659·22-s + 1.87·23-s − 0.467·24-s + 0.456·25-s − 0.332·26-s − 0.332·27-s − 0.474·29-s − 1.12·30-s − 0.807·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.377238710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.377238710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 11 | \( 1 - 0.309T + 11T^{2} \) |
| 13 | \( 1 - 1.69T + 13T^{2} \) |
| 17 | \( 1 - 4.15T + 17T^{2} \) |
| 19 | \( 1 + 6.09T + 19T^{2} \) |
| 23 | \( 1 - 8.96T + 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 + 4.49T + 31T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 - 7.97T + 41T^{2} \) |
| 43 | \( 1 - 5.37T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 5.07T + 61T^{2} \) |
| 67 | \( 1 - 5.30T + 67T^{2} \) |
| 71 | \( 1 - 4.27T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 6.07T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.949562340759198304182917752959, −7.41837520574261796399974978182, −6.57619022790696693571263807111, −5.92484528878051578741900200582, −5.21208207488087911959567846404, −4.04513298143256128106539549687, −3.25152439984822606956462135220, −2.48975433834742412975969188659, −1.91465998803896006694027552122, −0.995192530045798373006399477874,
0.995192530045798373006399477874, 1.91465998803896006694027552122, 2.48975433834742412975969188659, 3.25152439984822606956462135220, 4.04513298143256128106539549687, 5.21208207488087911959567846404, 5.92484528878051578741900200582, 6.57619022790696693571263807111, 7.41837520574261796399974978182, 7.949562340759198304182917752959
