L(s) = 1 | − 0.647·2-s − 0.415·3-s − 1.58·4-s − 5-s + 0.269·6-s + 7-s + 2.31·8-s − 2.82·9-s + 0.647·10-s − 1.53·11-s + 0.657·12-s − 0.279·13-s − 0.647·14-s + 0.415·15-s + 1.66·16-s + 3.36·17-s + 1.82·18-s − 0.328·19-s + 1.58·20-s − 0.415·21-s + 0.992·22-s + 1.30·23-s − 0.963·24-s + 25-s + 0.181·26-s + 2.42·27-s − 1.58·28-s + ⋯ |
L(s) = 1 | − 0.457·2-s − 0.240·3-s − 0.790·4-s − 0.447·5-s + 0.109·6-s + 0.377·7-s + 0.819·8-s − 0.942·9-s + 0.204·10-s − 0.462·11-s + 0.189·12-s − 0.0775·13-s − 0.172·14-s + 0.107·15-s + 0.415·16-s + 0.815·17-s + 0.431·18-s − 0.0752·19-s + 0.353·20-s − 0.0907·21-s + 0.211·22-s + 0.272·23-s − 0.196·24-s + 0.200·25-s + 0.0355·26-s + 0.466·27-s − 0.298·28-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 | \( 1 + 0.647T + 2T^{2} \) |
| 3 | \( 1 + 0.415T + 3T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 + 0.279T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 19 | \( 1 + 0.328T + 19T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 + 0.586T + 37T^{2} \) |
| 41 | \( 1 - 5.90T + 41T^{2} \) |
| 43 | \( 1 - 3.99T + 43T^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 53 | \( 1 + 0.955T + 53T^{2} \) |
| 59 | \( 1 - 0.727T + 59T^{2} \) |
| 61 | \( 1 + 4.22T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 3.93T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 + 9.64T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 7.40T + 89T^{2} \) |
| 97 | \( 1 + 7.42T + 97T^{2} \) |
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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
−7.75393844625271419947079270603, −7.03285562306779251217854692206, −5.88200989978891871582427886965, −5.38877703391824338603209491437, −4.78014117068843923202335282614, −3.89082736834066070974030924465, −3.22149130072221882405432058601, −2.11712729452224673122643044061, −0.939879593719348312106198045481, 0,
0.939879593719348312106198045481, 2.11712729452224673122643044061, 3.22149130072221882405432058601, 3.89082736834066070974030924465, 4.78014117068843923202335282614, 5.38877703391824338603209491437, 5.88200989978891871582427886965, 7.03285562306779251217854692206, 7.75393844625271419947079270603