L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 1.19·7-s − 8-s + 9-s − 10-s + 0.554·11-s − 12-s − 1.19·14-s − 15-s + 16-s − 6.45·17-s − 18-s − 5.40·19-s + 20-s − 1.19·21-s − 0.554·22-s + 7.96·23-s + 24-s + 25-s − 27-s + 1.19·28-s − 6.89·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.452·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.167·11-s − 0.288·12-s − 0.320·14-s − 0.258·15-s + 0.250·16-s − 1.56·17-s − 0.235·18-s − 1.24·19-s + 0.223·20-s − 0.261·21-s − 0.118·22-s + 1.65·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.226·28-s − 1.28·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.100531385\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.100531385\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 - 0.554T + 11T^{2} \) |
| 17 | \( 1 + 6.45T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 - 7.96T + 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 + 7.96T + 37T^{2} \) |
| 41 | \( 1 - 3.08T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 1.45T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 + 3.34T + 67T^{2} \) |
| 71 | \( 1 - 2.35T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 0.990T + 79T^{2} \) |
| 83 | \( 1 - 5.88T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−8.419911981977700624554963753905, −7.39435251571733651167964716335, −6.82166034402038784341262868804, −6.25221180093788425512539656800, −5.37834810517168410600239602866, −4.67282761443468632564715306582, −3.79017163757963711311596710788, −2.47041256071497167141201274386, −1.82479984396048362744413212778, −0.65148629031827451791568146291,
0.65148629031827451791568146291, 1.82479984396048362744413212778, 2.47041256071497167141201274386, 3.79017163757963711311596710788, 4.67282761443468632564715306582, 5.37834810517168410600239602866, 6.25221180093788425512539656800, 6.82166034402038784341262868804, 7.39435251571733651167964716335, 8.419911981977700624554963753905