| L(s) = 1 | + 2.69·2-s − 3-s + 5.28·4-s + 0.294·5-s − 2.69·6-s + 8.86·8-s + 9-s + 0.795·10-s + 4.64·11-s − 5.28·12-s − 3.49·13-s − 0.294·15-s + 13.3·16-s + 0.238·17-s + 2.69·18-s + 5.04·19-s + 1.55·20-s + 12.5·22-s + 1.01·23-s − 8.86·24-s − 4.91·25-s − 9.43·26-s − 27-s + 2.40·29-s − 0.795·30-s + 2.25·31-s + 18.3·32-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.577·3-s + 2.64·4-s + 0.131·5-s − 1.10·6-s + 3.13·8-s + 0.333·9-s + 0.251·10-s + 1.39·11-s − 1.52·12-s − 0.969·13-s − 0.0760·15-s + 3.34·16-s + 0.0577·17-s + 0.636·18-s + 1.15·19-s + 0.348·20-s + 2.67·22-s + 0.211·23-s − 1.81·24-s − 0.982·25-s − 1.85·26-s − 0.192·27-s + 0.447·29-s − 0.145·30-s + 0.405·31-s + 3.24·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.535909998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.535909998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 - 0.294T + 5T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 - 0.238T + 17T^{2} \) |
| 19 | \( 1 - 5.04T + 19T^{2} \) |
| 23 | \( 1 - 1.01T + 23T^{2} \) |
| 29 | \( 1 - 2.40T + 29T^{2} \) |
| 31 | \( 1 - 2.25T + 31T^{2} \) |
| 37 | \( 1 - 0.441T + 37T^{2} \) |
| 41 | \( 1 + 7.90T + 41T^{2} \) |
| 43 | \( 1 - 3.71T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 + 0.784T + 59T^{2} \) |
| 67 | \( 1 - 5.33T + 67T^{2} \) |
| 71 | \( 1 - 7.47T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 0.254T + 79T^{2} \) |
| 83 | \( 1 + 0.717T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 - 6.21T + 97T^{2} \) |
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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
−7.33533461680856022722178590637, −6.70971158570449185927953480476, −6.28118663060340941863734394691, −5.45668647573380899585186369775, −5.06631396929081349265668973133, −4.24656170021408738931532226179, −3.73482212389285925822255682773, −2.88634949024109323411241892855, −2.01074325833242265058562121894, −1.09180922624343499655871746923,
1.09180922624343499655871746923, 2.01074325833242265058562121894, 2.88634949024109323411241892855, 3.73482212389285925822255682773, 4.24656170021408738931532226179, 5.06631396929081349265668973133, 5.45668647573380899585186369775, 6.28118663060340941863734394691, 6.70971158570449185927953480476, 7.33533461680856022722178590637
