L(s) = 1 | − 0.583·2-s − 0.502·3-s − 1.65·4-s + 5-s + 0.293·6-s − 7-s + 2.13·8-s − 2.74·9-s − 0.583·10-s − 3.81·11-s + 0.833·12-s + 3.99·13-s + 0.583·14-s − 0.502·15-s + 2.07·16-s − 6.61·17-s + 1.60·18-s + 1.33·19-s − 1.65·20-s + 0.502·21-s + 2.22·22-s + 2.07·23-s − 1.07·24-s + 25-s − 2.32·26-s + 2.88·27-s + 1.65·28-s + ⋯ |
L(s) = 1 | − 0.412·2-s − 0.290·3-s − 0.829·4-s + 0.447·5-s + 0.119·6-s − 0.377·7-s + 0.754·8-s − 0.915·9-s − 0.184·10-s − 1.14·11-s + 0.240·12-s + 1.10·13-s + 0.155·14-s − 0.129·15-s + 0.518·16-s − 1.60·17-s + 0.377·18-s + 0.306·19-s − 0.371·20-s + 0.109·21-s + 0.474·22-s + 0.432·23-s − 0.218·24-s + 0.200·25-s − 0.456·26-s + 0.555·27-s + 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 + 0.583T + 2T^{2} \) |
| 3 | \( 1 + 0.502T + 3T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 19 | \( 1 - 1.33T + 19T^{2} \) |
| 23 | \( 1 - 2.07T + 23T^{2} \) |
| 29 | \( 1 - 3.04T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 - 6.69T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 - 6.08T + 43T^{2} \) |
| 47 | \( 1 + 5.97T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 7.05T + 59T^{2} \) |
| 61 | \( 1 - 0.224T + 61T^{2} \) |
| 67 | \( 1 + 7.32T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 0.695T + 83T^{2} \) |
| 89 | \( 1 - 2.16T + 89T^{2} \) |
| 97 | \( 1 - 0.603T + 97T^{2} \) |
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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
−7.64558852958140479470451163366, −6.71023621561226041539325493985, −6.06832205127165387135473049862, −5.38720382291659880778724647631, −4.81513739955133819927076372159, −3.95632759034440629721603160426, −3.02835309541082628045756636872, −2.23448738826999733891737355020, −0.952778771109724133781208308287, 0,
0.952778771109724133781208308287, 2.23448738826999733891737355020, 3.02835309541082628045756636872, 3.95632759034440629721603160426, 4.81513739955133819927076372159, 5.38720382291659880778724647631, 6.06832205127165387135473049862, 6.71023621561226041539325493985, 7.64558852958140479470451163366