L(s) = 1 | − 2-s − 3-s + 4-s − 0.664·5-s + 6-s + 2.36·7-s − 8-s + 9-s + 0.664·10-s − 3.14·11-s − 12-s + 13-s − 2.36·14-s + 0.664·15-s + 16-s − 7.53·17-s − 18-s + 5.47·19-s − 0.664·20-s − 2.36·21-s + 3.14·22-s + 1.27·23-s + 24-s − 4.55·25-s − 26-s − 27-s + 2.36·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.297·5-s + 0.408·6-s + 0.892·7-s − 0.353·8-s + 0.333·9-s + 0.210·10-s − 0.947·11-s − 0.288·12-s + 0.277·13-s − 0.631·14-s + 0.171·15-s + 0.250·16-s − 1.82·17-s − 0.235·18-s + 1.25·19-s − 0.148·20-s − 0.515·21-s + 0.670·22-s + 0.266·23-s + 0.204·24-s − 0.911·25-s − 0.196·26-s − 0.192·27-s + 0.446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8666638931\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8666638931\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 0.664T + 5T^{2} \) |
| 7 | \( 1 - 2.36T + 7T^{2} \) |
| 11 | \( 1 + 3.14T + 11T^{2} \) |
| 17 | \( 1 + 7.53T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 29 | \( 1 + 5.06T + 29T^{2} \) |
| 31 | \( 1 - 6.27T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 6.25T + 43T^{2} \) |
| 47 | \( 1 + 8.67T + 47T^{2} \) |
| 53 | \( 1 + 9.70T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 2.60T + 61T^{2} \) |
| 67 | \( 1 + 6.78T + 67T^{2} \) |
| 71 | \( 1 - 1.70T + 71T^{2} \) |
| 73 | \( 1 - 0.395T + 73T^{2} \) |
| 79 | \( 1 - 2.79T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 9.21T + 97T^{2} \) |
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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
−7.71943458955861037770012450386, −7.50028018863998042086037166447, −6.35950269194257831810965489756, −5.98752603668921784137721548199, −4.83163968407724308237504792013, −4.64271952691743709289686539361, −3.42531973668065294270936306773, −2.44088403196339337652958017057, −1.62662402491856799271559236727, −0.53278370470825405966474709112,
0.53278370470825405966474709112, 1.62662402491856799271559236727, 2.44088403196339337652958017057, 3.42531973668065294270936306773, 4.64271952691743709289686539361, 4.83163968407724308237504792013, 5.98752603668921784137721548199, 6.35950269194257831810965489756, 7.50028018863998042086037166447, 7.71943458955861037770012450386