| L(s) = 1 | − 1.60·2-s + 2.74·3-s + 0.571·4-s − 5-s − 4.40·6-s + 7-s + 2.29·8-s + 4.54·9-s + 1.60·10-s − 2.96·11-s + 1.57·12-s − 0.939·13-s − 1.60·14-s − 2.74·15-s − 4.81·16-s − 8.10·17-s − 7.29·18-s + 3.13·19-s − 0.571·20-s + 2.74·21-s + 4.74·22-s + 5.95·23-s + 6.29·24-s + 25-s + 1.50·26-s + 4.25·27-s + 0.571·28-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 1.58·3-s + 0.285·4-s − 0.447·5-s − 1.79·6-s + 0.377·7-s + 0.809·8-s + 1.51·9-s + 0.507·10-s − 0.892·11-s + 0.453·12-s − 0.260·13-s − 0.428·14-s − 0.709·15-s − 1.20·16-s − 1.96·17-s − 1.71·18-s + 0.718·19-s − 0.127·20-s + 0.599·21-s + 1.01·22-s + 1.24·23-s + 1.28·24-s + 0.200·25-s + 0.295·26-s + 0.818·27-s + 0.108·28-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 | \( 1 + 1.60T + 2T^{2} \) |
| 3 | \( 1 - 2.74T + 3T^{2} \) |
| 11 | \( 1 + 2.96T + 11T^{2} \) |
| 13 | \( 1 + 0.939T + 13T^{2} \) |
| 17 | \( 1 + 8.10T + 17T^{2} \) |
| 19 | \( 1 - 3.13T + 19T^{2} \) |
| 23 | \( 1 - 5.95T + 23T^{2} \) |
| 29 | \( 1 - 1.13T + 29T^{2} \) |
| 31 | \( 1 - 2.81T + 31T^{2} \) |
| 37 | \( 1 + 5.15T + 37T^{2} \) |
| 41 | \( 1 - 8.49T + 41T^{2} \) |
| 43 | \( 1 + 7.36T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 4.01T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 + 4.48T + 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 + 2.52T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 5.40T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−7.69966746556420975737513440337, −7.26031821600766824608401727013, −6.60420325705379838527008679793, −5.05991695067168506484513060075, −4.63036744675159432906283854989, −3.75466149921047716528671343792, −2.81472497878897633981985713862, −2.23963506579344615805856263903, −1.29978597849727576244550348881, 0,
1.29978597849727576244550348881, 2.23963506579344615805856263903, 2.81472497878897633981985713862, 3.75466149921047716528671343792, 4.63036744675159432906283854989, 5.05991695067168506484513060075, 6.60420325705379838527008679793, 7.26031821600766824608401727013, 7.69966746556420975737513440337
