| L(s) = 1 | − 4.93·2-s − 0.435·3-s + 16.3·4-s − 4.99·5-s + 2.14·6-s − 7·7-s − 41.2·8-s − 26.8·9-s + 24.6·10-s − 23.2·11-s − 7.11·12-s + 9.93·13-s + 34.5·14-s + 2.17·15-s + 72.6·16-s + 132.·18-s − 74.4·19-s − 81.7·20-s + 3.04·21-s + 114.·22-s − 45.6·23-s + 17.9·24-s − 100.·25-s − 49.0·26-s + 23.4·27-s − 114.·28-s + 112.·29-s + ⋯ |
| L(s) = 1 | − 1.74·2-s − 0.0837·3-s + 2.04·4-s − 0.447·5-s + 0.146·6-s − 0.377·7-s − 1.82·8-s − 0.992·9-s + 0.780·10-s − 0.638·11-s − 0.171·12-s + 0.211·13-s + 0.659·14-s + 0.0374·15-s + 1.13·16-s + 1.73·18-s − 0.899·19-s − 0.914·20-s + 0.0316·21-s + 1.11·22-s − 0.413·23-s + 0.152·24-s − 0.800·25-s − 0.369·26-s + 0.166·27-s − 0.772·28-s + 0.722·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.001013884655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001013884655\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 4.93T + 8T^{2} \) |
| 3 | \( 1 + 0.435T + 27T^{2} \) |
| 5 | \( 1 + 4.99T + 125T^{2} \) |
| 11 | \( 1 + 23.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.93T + 2.19e3T^{2} \) |
| 19 | \( 1 + 74.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 56.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 84.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 444.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 746.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 224.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 575.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 135.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 177.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.12e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 179.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 399.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 121.T + 9.12e5T^{2} \) |
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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
−8.744351305650396714948354525581, −8.098198136803993185596103166431, −7.65536186878158979611222520944, −6.56230450474325391629608522961, −6.06710409853188900059396246004, −4.85135736275613768337587538279, −3.47204300719338520075948871067, −2.57913371910045129426055090789, −1.57038275710336050788770131009, −0.01873685443208940737767061600,
0.01873685443208940737767061600, 1.57038275710336050788770131009, 2.57913371910045129426055090789, 3.47204300719338520075948871067, 4.85135736275613768337587538279, 6.06710409853188900059396246004, 6.56230450474325391629608522961, 7.65536186878158979611222520944, 8.098198136803993185596103166431, 8.744351305650396714948354525581
