| L(s) = 1 | + 2-s + 3.02·3-s + 4-s + 3.40·5-s + 3.02·6-s − 2.58·7-s + 8-s + 6.14·9-s + 3.40·10-s + 0.981·11-s + 3.02·12-s + 5.52·13-s − 2.58·14-s + 10.2·15-s + 16-s + 2.66·17-s + 6.14·18-s − 0.439·19-s + 3.40·20-s − 7.81·21-s + 0.981·22-s − 23-s + 3.02·24-s + 6.58·25-s + 5.52·26-s + 9.49·27-s − 2.58·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.74·3-s + 0.5·4-s + 1.52·5-s + 1.23·6-s − 0.977·7-s + 0.353·8-s + 2.04·9-s + 1.07·10-s + 0.296·11-s + 0.872·12-s + 1.53·13-s − 0.690·14-s + 2.65·15-s + 0.250·16-s + 0.645·17-s + 1.44·18-s − 0.100·19-s + 0.761·20-s − 1.70·21-s + 0.209·22-s − 0.208·23-s + 0.617·24-s + 1.31·25-s + 1.08·26-s + 1.82·27-s − 0.488·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.158584081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.158584081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 3.02T + 3T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 7 | \( 1 + 2.58T + 7T^{2} \) |
| 11 | \( 1 - 0.981T + 11T^{2} \) |
| 13 | \( 1 - 5.52T + 13T^{2} \) |
| 17 | \( 1 - 2.66T + 17T^{2} \) |
| 19 | \( 1 + 0.439T + 19T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 + 0.467T + 31T^{2} \) |
| 37 | \( 1 + 2.01T + 37T^{2} \) |
| 41 | \( 1 + 5.82T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 + 5.21T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 6.13T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 + 8.69T + 73T^{2} \) |
| 79 | \( 1 + 5.26T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 3.90T + 97T^{2} \) |
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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
−8.264178094219701084210906669743, −7.25926366417227820472901967107, −6.56003817797538341933404141141, −6.03782005695900342679570418526, −5.27802611317731078482440614638, −4.12562678586468839925656870755, −3.30322787150965307165980246345, −3.12685566103475241854735638706, −1.89383136320122055081581155055, −1.56017891247795615982458235646,
1.56017891247795615982458235646, 1.89383136320122055081581155055, 3.12685566103475241854735638706, 3.30322787150965307165980246345, 4.12562678586468839925656870755, 5.27802611317731078482440614638, 6.03782005695900342679570418526, 6.56003817797538341933404141141, 7.25926366417227820472901967107, 8.264178094219701084210906669743
