| L(s) = 1 | − 2-s − 0.191·3-s + 4-s − 1.60·5-s + 0.191·6-s + 3.22·7-s − 8-s − 2.96·9-s + 1.60·10-s − 3.96·11-s − 0.191·12-s + 1.74·13-s − 3.22·14-s + 0.307·15-s + 16-s − 6.02·17-s + 2.96·18-s − 2.71·19-s − 1.60·20-s − 0.618·21-s + 3.96·22-s − 7.74·23-s + 0.191·24-s − 2.42·25-s − 1.74·26-s + 1.14·27-s + 3.22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.110·3-s + 0.5·4-s − 0.718·5-s + 0.0782·6-s + 1.21·7-s − 0.353·8-s − 0.987·9-s + 0.507·10-s − 1.19·11-s − 0.0553·12-s + 0.483·13-s − 0.861·14-s + 0.0794·15-s + 0.250·16-s − 1.46·17-s + 0.698·18-s − 0.622·19-s − 0.359·20-s − 0.134·21-s + 0.844·22-s − 1.61·23-s + 0.0391·24-s − 0.484·25-s − 0.341·26-s + 0.220·27-s + 0.609·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6662707675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6662707675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 0.191T + 3T^{2} \) |
| 5 | \( 1 + 1.60T + 5T^{2} \) |
| 7 | \( 1 - 3.22T + 7T^{2} \) |
| 11 | \( 1 + 3.96T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 + 6.02T + 17T^{2} \) |
| 19 | \( 1 + 2.71T + 19T^{2} \) |
| 23 | \( 1 + 7.74T + 23T^{2} \) |
| 29 | \( 1 + 0.940T + 29T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 - 8.86T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 - 6.72T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 2.99T + 59T^{2} \) |
| 61 | \( 1 + 0.0148T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 6.43T + 73T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 + 3.66T + 83T^{2} \) |
| 89 | \( 1 + 2.39T + 89T^{2} \) |
| 97 | \( 1 - 1.29T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.478173652922323107805743151194, −7.908510781652612410746840078402, −7.29147472314276459660751495543, −6.19237144666459826632911583437, −5.62245965075213669872531812067, −4.60246428306355499508812994513, −3.95257558985850012669026937153, −2.60196785020696506944831019813, −2.03955997723006353113282587925, −0.49547091390334713649317348590,
0.49547091390334713649317348590, 2.03955997723006353113282587925, 2.60196785020696506944831019813, 3.95257558985850012669026937153, 4.60246428306355499508812994513, 5.62245965075213669872531812067, 6.19237144666459826632911583437, 7.29147472314276459660751495543, 7.908510781652612410746840078402, 8.478173652922323107805743151194
