L(s) = 1 | − 2-s + 3-s + 4-s + 0.290·5-s − 6-s − 3.15·7-s − 8-s + 9-s − 0.290·10-s + 5.57·11-s + 12-s − 13-s + 3.15·14-s + 0.290·15-s + 16-s − 2.13·17-s − 18-s − 5.05·19-s + 0.290·20-s − 3.15·21-s − 5.57·22-s + 1.96·23-s − 24-s − 4.91·25-s + 26-s + 27-s − 3.15·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.129·5-s − 0.408·6-s − 1.19·7-s − 0.353·8-s + 0.333·9-s − 0.0919·10-s + 1.68·11-s + 0.288·12-s − 0.277·13-s + 0.843·14-s + 0.0750·15-s + 0.250·16-s − 0.518·17-s − 0.235·18-s − 1.15·19-s + 0.0649·20-s − 0.688·21-s − 1.18·22-s + 0.410·23-s − 0.204·24-s − 0.983·25-s + 0.196·26-s + 0.192·27-s − 0.596·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 0.290T + 5T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 - 5.57T + 11T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 + 5.05T + 19T^{2} \) |
| 23 | \( 1 - 1.96T + 23T^{2} \) |
| 29 | \( 1 + 0.444T + 29T^{2} \) |
| 31 | \( 1 - 2.60T + 31T^{2} \) |
| 37 | \( 1 + 7.55T + 37T^{2} \) |
| 41 | \( 1 - 8.43T + 41T^{2} \) |
| 43 | \( 1 + 1.60T + 43T^{2} \) |
| 47 | \( 1 - 9.45T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 0.462T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 8.17T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 7.96T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−7.37016470354776054078188769891, −6.90630667279814734011737530929, −6.31826604185087046974613137238, −5.72848577214127904096579297700, −4.30217661990212031993868702600, −3.89245405377752096247136978793, −2.95596773388151833021471189298, −2.19917920367049794069490500641, −1.26544681117673891306055167251, 0,
1.26544681117673891306055167251, 2.19917920367049794069490500641, 2.95596773388151833021471189298, 3.89245405377752096247136978793, 4.30217661990212031993868702600, 5.72848577214127904096579297700, 6.31826604185087046974613137238, 6.90630667279814734011737530929, 7.37016470354776054078188769891