L(s) = 1 | − 2-s − 1.47·3-s + 4-s + 5-s + 1.47·6-s − 2.32·7-s − 8-s − 0.820·9-s − 10-s + 1.34·11-s − 1.47·12-s + 3.18·13-s + 2.32·14-s − 1.47·15-s + 16-s + 3.91·17-s + 0.820·18-s − 4.40·19-s + 20-s + 3.42·21-s − 1.34·22-s − 4.06·23-s + 1.47·24-s + 25-s − 3.18·26-s + 5.64·27-s − 2.32·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.852·3-s + 0.5·4-s + 0.447·5-s + 0.602·6-s − 0.877·7-s − 0.353·8-s − 0.273·9-s − 0.316·10-s + 0.405·11-s − 0.426·12-s + 0.883·13-s + 0.620·14-s − 0.381·15-s + 0.250·16-s + 0.950·17-s + 0.193·18-s − 1.01·19-s + 0.223·20-s + 0.748·21-s − 0.286·22-s − 0.847·23-s + 0.301·24-s + 0.200·25-s − 0.624·26-s + 1.08·27-s − 0.438·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 1.47T + 3T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 - 3.18T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 + 4.40T + 19T^{2} \) |
| 23 | \( 1 + 4.06T + 23T^{2} \) |
| 29 | \( 1 + 8.96T + 29T^{2} \) |
| 31 | \( 1 - 1.15T + 31T^{2} \) |
| 37 | \( 1 - 0.657T + 37T^{2} \) |
| 41 | \( 1 - 7.76T + 41T^{2} \) |
| 43 | \( 1 - 5.75T + 43T^{2} \) |
| 47 | \( 1 + 7.10T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 + 2.03T + 59T^{2} \) |
| 61 | \( 1 - 1.33T + 61T^{2} \) |
| 67 | \( 1 + 1.04T + 67T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 73 | \( 1 + 7.58T + 73T^{2} \) |
| 79 | \( 1 + 8.46T + 79T^{2} \) |
| 83 | \( 1 + 3.04T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.182739465555414739566569148664, −7.28248627824577299927325372400, −6.44800203273882545820699310892, −5.95932852399261621235367285258, −5.54564825269203541495685432034, −4.19099668426758454747983181104, −3.35314274589004306032885178822, −2.29497988778835994437268230440, −1.14497977190670319467270237010, 0,
1.14497977190670319467270237010, 2.29497988778835994437268230440, 3.35314274589004306032885178822, 4.19099668426758454747983181104, 5.54564825269203541495685432034, 5.95932852399261621235367285258, 6.44800203273882545820699310892, 7.28248627824577299927325372400, 8.182739465555414739566569148664